Abstract
Let X be a compact Kähler manifold. Let \(T_1, \ldots , T_m\) be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of \(T_1, \ldots , T_m\). We recover the well-known non-pluripolar product of \(T_1, \ldots , T_m\) when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.
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1 Introduction
The problem of defining the intersection of closed positive currents on a complex manifold is a central question in the pluripotential theory. This question is already important for currents of bi-degree (1, 1) with deep applications to complex geometry as well as complex dynamics; see [1, 11, 15, 16, 18] and references therein for an introduction to the subject.
Let X be an arbitrary complex manifold. Let \(T_1, \ldots ,T_m\) be closed positive currents of bi-degree (1, 1) on X. In [2, 4, 17], the non-pluripolar product \(\langle T_1 \wedge \cdots \wedge T_m\rangle \) of \(T_1, \ldots , T_m\) was defined by using the quasi-continuity of plurisubharmonic (psh for short) functions with respect to capacity. Since then, this notion has played an important role in complex geometry. We refer to [7,8,9,10] for recent developments.
In this paper, our first goal is to generalize the notion of non-pluripolar product to a more general natural setting. Given a closed positive (p, p)-current T on X, we introduce the non-pluripolar product relative to T of \(T_1, \ldots , T_m\) which we denote by \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \). We recover the above-mentioned non-pluripolar product when T is the current of integration along X (see also Remark 3.8); and if \(T_1, \ldots , T_m\) are of locally bounded potentials, then \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is nothing but the classical intersection of \(T_1, \ldots , T_m,T\). When well defined, \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is a closed positive \((p+m,p+m)\)-current on X. As in [4], the proof of the last fact follows from ideas in [21]. The product \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is symmetric, homogeneous and sub-additive in \(T_1, \ldots , T_m\). One can consider the last product as a sort of intersection of \(T_1, \ldots , T_m,T\).
The main ingredient in the construction of relative non-pluripolar products is a (uniform) quasi-continuity property of bounded psh functions with respect to the capacity associated to closed positive currents (Theorem 2.4). This quasi-continuity property is stronger than the usual one. We also need to establish some convergence properties of mixed Monge–Ampère operators which are of independent interest.
Consider now the case where X is a compact Kähler manifold. As one can expect, the relative non-pluripolar products are always well defined in this setting. For any closed positive current R in X, we denote by \(\{R\}\) the cohomology class of R. For cohomology classes \(\alpha ,\beta \) in \(H^{p,p}(X,{\mathbb {R}})\), we write \(\alpha \le \beta \) if \((\beta -\alpha )\) can be represented by a closed positive current. The following result gives a monotonicity property of relative non-pluripolar products.
Theorem 1.1
(Theorem 4.4) Let X be a compact Kähler manifold and \(T_1,\ldots , T_m,T\) closed positive currents on X such that \(T_j\) is of bi-degree (1, 1) for \(1 \le j \le m\). Let \(T'_j\) be closed positive (1, 1)-current in the cohomology class of \(T_j\)on Xsuch that \(T'_j\)is less singular than \(T_j\)for \(1 \le j \le m\). Then we have
Note that even when T is the current of integration along X, this result is new. In the last case, the above result was conjectured in [4] and proved there for \(T_j\) of potentials locally bounded outside a closed complete pluripolar set. The last condition was relaxed in [9, 22] but it was required there that \(m=\dim X\) (always for T to be the current of integration along X). The proofs of these results presented there do not extend to our setting. A key ingredient in our proof is Theorem 2.6 (and Remark 2.7) giving a generalization of well-known convergence properties of Monge–Ampère operators. To prove Theorem 2.6, we will need the strong quasi-continuity of bounded psh functions mentioned above. When T is of bi-degree (1, 1), we actually have a stronger monotonicity property, see Remark 4.5.
Theorem 1.1 allows us to define the notion of full mass intersection relative to T for currents \(T_1, \ldots , T_m\) as in [4], see Definition 4.7. Our next goal is to study currents of relative full mass intersection. We will prove that the relative non-pluripolar products of currents with full mass intersection are continuous under decreasing or increasing sequences (Theorem 4.8).
Now we concentrate on the case where the cohomology classes of \(T_1, \ldots , T_m\) are Kähler. In this case, \(T_1, \ldots , T_m\) are of full mass intersection relative to T if and only if \(\{ \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T\rangle \}\) is equal to \(\{\bigwedge _{j=1}^m \theta _j \wedge T\}\), where \(\theta _j\) is a closed smooth (1, 1)-form cohomologous to \(T_j\) for \(1 \le j \le m\). Our next main result explains why having positive Lelong numbers is an obstruction for currents to be of full mass intersection.
Theorem 1.2
(Theorem 4.11) Let Xbe a compact Kähler manifold. Let \(T_1, \ldots , T_m\) be closed positive (1, 1)-currents on X such that the cohomology class of \(T_j\) is Kähler for \(1 \le j \le m\) and T a closed positive (p, p)-current on X with \(p+m \le n\). Let V be an irreducible analytic subset in X such that the generic Lelong numbers of \(T_1, \ldots , T_m,T\) along V are strictly positive. Assume \(T_1, \ldots , T_m\) are of full mass intersection relative to T. Then, we have \(\dim V <n-p-m\).
Recall that the generic Lelong number of T along V is the smallest value among the Lelong numbers of T at points in V. When \(p+m=n\), the above theorem says that V is empty. To get motivated about Theorem 1.2, let us consider the case where \(p+m=n\) and the current \(T_j\) has a potential which is locally bounded outside a point \(x_0\) in X. In this case, \(T_1 \wedge \cdots \wedge T_m \wedge T\) is well defined in the sense given in [1, 11, 16] and we can see that \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle = T_1 \wedge \cdots \wedge T_m \wedge T\) on \(X \backslash \{x_0\}\). The latter current has strictly positive mass on \(x_0\) by [11, Corollary 7.9]. Hence, \(T_1, \ldots , T_m\) cannot be of full mass intersection relative to T. By considering \(T_j\) to be suitable currents with analytic singularities (for example when X is projective) and T is the current of integration along X, one can see that the conclusion of Theorem 1.2 is optimal.
When \(m=n\) and \(T_1= \cdots = T_m\), Theorem 1.2 was proved in [17]. Their proof uses comparison principle and hence does not apply to our setting because \(T_1, \ldots , T_m\) are different in general. When the class of \(T_j\) is not Kähler, the above theorem no longer holds because currents with minimal singularities in a big and non-nef cohomology class always have positive Lelong numbers at some points in X (see [3]). The proof of Theorem 1.2, which is Theorem 4.11 below, uses Theorem 1.1.
Let \({\mathcal {W}}^-\) be the set of convex increasing functions \(\chi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) with \(\chi (-\infty )= -\infty \). A function in \({\mathcal {W}}^{-}\) is called a (convex) weight. We will define the notion of having full mass intersection relative to T with weight \(\chi \) and the weighted class of these currents, see Definition 5.3. When T is the current of integration along X and \(T_1, \ldots , T_m\) are all equal, the notion of weighted class (or pluricomplex energy) was introduced in [6] in the local context. Analogous global notions were later studied in [4, 17] and in other subsequent works. Here is our main result in this part.
Theorem 1.3
Let \(X, T_1,\ldots , T_m,T\) be as in Theorem 1.2. Let \(\chi \in {\mathcal {W}}^-\). The following three assertions hold:
-
(i)
If \(T_j, \ldots , T_j\) (m times \(T_j\)) are of full mass intersection relative to T for \(1 \le j \le m\), then \(T_1, \ldots , T_m\) are also of full mass intersection relative to T.
-
(ii)
Let \(T'_j \) be a closed positive(1, 1)-current whose cohomology class is Kähler for\(1 \le j \le m\). Assume that \(T'_j\) is less singular than\(T_j\) and \(T_1, \ldots , T_m\) are of full mass intersection relative to T with weight \(\chi \). Then \(T'_1, \ldots , T'_m\) are also of full mass intersection relative to T with weight \(\chi \).
-
(iii)
If \(T_j, \ldots , T_j\) (m times \(T_j\)) are of full mass intersection relative to T with weight \(\chi \) for \(1 \le j \le m\), then \(T_1, \ldots , T_m\) are also of full mass intersection relative to T with weight \(\chi \).
Theorem 1.3 is a combination of Theorems 5.1, 5.8 and 5.9 below. Property \((i)-(ii)\) of Theorem 1.3 was proved in [17] in the case where T is the current of integration along X and \(T_1, \ldots , T_m\) are equal and \(T'_1, \ldots , T'_m\) are equal. When T is the current of integration along X, Property (iii) of Theorem 1.3 was proved in [17] for \(\chi (t)= -(-t)^r\) with \(0< r \le 1\) (see also the last paper and [6] for the case where \(r\ge 1\)) and it was proved in [7] for every \(\chi \in {\mathcal {W}}^-\) under an extra condition that \(m=n\). The proof given in [7] used Monge–Ampère equations, hence, does not extend to our setting.
We have some comments about the proof of Theorem 1.3. Firstly, arguments similar to those in [17] are sufficient to obtain (ii) of Theorem 1.3. However, as pointed out above, to prove (iii), we need new arguments. We prove (i), (iii) using the same approach. It is possible that we can use the comparison principle to get (i) as in the case where T is the current of integration along X. But we choose to invoke a more flexible idea which is also applicable to get (iii). The idea is to combine a monotonicity property and generalizations of results in [2] about plurifine topology properties of Monge–Ampère operators. Here, the monotonicity mentioned in the last sentence is Theorem 1.1 in case of proving (i) and (ii) in case of proving (iii).
The paper is organized as follows. In Sect. 2, we establish the uniform strong quasi-continuity of bounded psh functions and derive from it some consequences concerning the convergence of Monge–Ampère operators. In Sect. 3, we define the notion of relative non-pluripolar products and prove its basic properties. In Sect. 4, we prove the monotonicity of relative non-pluripolar products and explain Lelong number obstruction for currents having full mass intersection. Section 5 is devoted to the weighted class of currents with relative full mass intersection.
Notation and convention. For a closed positive current T on a compact complex manifold X, we denote by \(\{T\}\) the cohomology class of T and \(\nu (T,x)\) denotes the Lelong number of T at x. Recall that the wedge product on closed smooth forms induces the cup-product on their de Rham cohomology classes. We use the same notation \(`` \wedge "\) to denote these two products.
Recall \(\mathrm{d}^c= \frac{i}{2\pi } ({\overline{\partial }} - \partial )\) and \(\mathrm{dd}^c= \frac{i}{\pi } \partial {\overline{\partial }}\). We use \(\gtrsim , \lesssim \) to denote \(\ge , \le \) modulo some multiplicative constant independent of parameters in question. For a Borel set \(A \subset X\), we denote by \(\mathbf{1} _A\) the characteristic function of A, that means \(\mathbf{1} _A\) is equal to 1 on A and 0 elsewhere. For every current R of order 0 on X, we denote by \(\Vert R\Vert _A\) the mass of R on A.
2 Quasi-continuity for bounded psh functions
In this section, we prove a quasi-continuity for bounded plurisubharmonic (psh for short) functions which is stronger than the one for general psh functions. This property is the key to define our generalization of non-pluripolar products.
Let U be an open subset of \({\mathbb {C}}^n\). Let K be a Borel subset of U. The capacity \(\text {cap}(K,U)\) of K in U, which was introduced in [1], is given by
For a closed positive current T of bi-dimension (m, m) on U (\(0 \le m \le n\)), we define
We say that a sequence of functions \((u_k)_{k \in {\mathbb {N}}}\) converges to u with respect to the capacity \(\text {cap}_T\) (relative in U) if for any constant \(\epsilon >0\) and \(K \Subset U\), we have \(\text {cap}_T\big (\{|u_k - u| \ge \epsilon \}\cap K, U\big ) \rightarrow 0\) as \(k \rightarrow \infty \). The notion of relative \(\text {cap}_T\) was introduced in [18, 23]. We say that a subset A in U is locally complete pluripolar set if locally \(A= \{\psi = -\infty \}\) for some psh function \(\psi \). We begin with the following lemma which is probably well-known.
Lemma 2.1
Let A be a locally complete pluripolar set in U. Let T be a closed positive current of bi-dimension (m, m) on U. Assume that T has no mass on A. Then, we have \(\text {cap}_T(A,U)=0\).
Proof
The proof is standard. We present the details for readers’ convenience. Since the problem is of local nature, we can assume that there is a negative psh function \(\psi \) on U such that \(A= \{\psi = -\infty \}\). Let \(u_1,\ldots , u_m\) be bounded psh functions on U such that \(0 \le u_j \le 1\) for \(1 \le j \le m\). Let \(\omega \) is the standard Kähler form on \({\mathbb {C}}^n\). Let \(k \in {\mathbb {N}}\) and \(\psi _k:= k^{-1} \max \{\psi , -k\}\). We have \(-1 \le \psi _k \le 0\). Let \(\chi \) be a nonnegative smooth function with compact support in U. Let \(0 \le l \le m\) be an integer. Put
Since \(\psi _k =-1\) on \(\{\psi <-k\}\), in order to prove the desired assertion, it is enough to show that for every \(0 \le l \le q\), we have
as \(k \rightarrow \infty \) uniformly in \(u_1,\ldots , u_l\). We will prove (2.1) by induction on l. Firstly, (2.1) is trivial if \(l=0\) because T has no mass on A. Assume that it holds for \((l-1)\). We prove it for l. Put
By integration by parts, we have
Denote by \(I_{k,1}, I_{k,2}, I_{k,3}\) the first, second and third term, respectively, in the right-hand side of the last equality. Since \(u_1\) is bounded by 1, by integration by parts, we get
for some constant C depending only on \(\chi \). By induction hypothesis, we have
Thus \(\lim _{k \rightarrow \infty } I_{k,j} =0\) for \(j=1,2\). To treat \(I_{k,3}\), we use the Cauchy–Schwarz inequality to get
Let \(0 \le \chi _1 \le 1\) be a smooth cut-off function compactly supported on U such that \(\chi _1= 1\) on \(\mathrm{Supp}\chi \). Let \(U_1 \Subset U\) be an open subset containing \(\mathrm{Supp}\chi _1\). Since \(d \psi _k \wedge \mathrm{d}^c\psi _k \wedge R \ge 0\), we have
because \(-1 \le \psi _k \le 0\) and \(- \omega \lesssim \mathrm{dd}^c\chi _1 \lesssim \omega \). We infer that
By induction hypothesis, \(\lim _{k \rightarrow \infty }\int _{U_1} \psi _k R \wedge \omega = 0\). So \(\lim _{k\rightarrow \infty } I_{k,3}=0\). In conclusion, (2.1) follows. This finishes the proof. \(\square \)
We now give a definition which will be important later. Let \((T_k)_k\) be a sequence of closed positive currents of bi-dimension (m, m) on U. We say that \((T_k)_k\) satisfies Condition \((*)\) if \((T_k)_k\) is of uniformly bounded mass on compact subsets of U, and for every open set \(U' \subset U\) and every bounded psh function u on \(U'\) and every sequence \((u_k)_k\) of psh functions on \(U'\) decreasing to u, we have
An obvious example for sequences satisfying Condition \((*)\) is constant sequences: \(T_k= T\) for every k. We can also take \((T_k)_k\) to be a sequence of suitable Monge–Ampère operators with decreasing potentials, see [1, 11, 16]. Concerning Condition \((*)\), we will use mostly the example of constant sequences and the one provided by the following result.
Theorem 2.2
Let S be a closed positive current on U. Let v be a psh function on U such that v is locally integrable with respect to the trace measure of S and \((v_k)_k\) a sequence of psh functions on U such that \(v_k \rightarrow v\) in \(L^1_{loc}\) as \(k \rightarrow \infty \) and \(v_k \ge v\) for every k. Let \(T:= \mathrm{dd}^cv\wedge S\) and \(T_k:= \mathrm{dd}^cv_k \wedge S\). Let \(u_j\) be a bounded psh function on U for \(1 \le j \le m\). Let \((u_{jk})_{k \in {\mathbb {N}}}\) be a sequence of uniformly bounded psh functions such that \(u_{jk} \rightarrow u_j\) in \(L^1_{loc}\) as \(k \rightarrow \infty \) and \(u_{jk} \ge u_j\) for every j, k. Then we have
as \(k \rightarrow \infty \). In particular, the sequence \((T_k)_k\) satisfies Condition \((*)\).
Proof
By Hartog’s lemma, \(v_k, u_{jk}\) are uniformly bounded from above in k on compact subsets of U for every j. Since the problem is local, as usual, we can assume that U is relatively compact open set with smooth boundary in \({\mathbb {C}}^n\), every psh function in questions is defined on an open neighborhood of \({\overline{U}}\), \(v_k, v \le 0\) on U for every k and \(u_{jk}, u_j\) are all equal to a smooth psh function \(\psi \) outside some fixed compact subset of U such that \(\psi =0\) on \(\partial U\). We claim that
as \(k \rightarrow \infty \). In particular, this implies that v is locally integrable with respect to \(\mathrm{dd}^cu_{1} \wedge \cdots \wedge \mathrm{dd}^cu_{m}\wedge S\). We will prove (2.3) and (2.4) simultaneously by induction on m. When \(m=0\), this follows directly from [16, Proposition 3.2]. Assume that (2.3) and (2.4) hold for \((m-1)\) in place of m. Let
for \(1 \le j \le m\). By induction hypothesis, we have
for \(j \ge 2\). Since \(u_{1k}\) is uniformly bounded on U, the family \(u_{1k} R_{2,k}\) is of uniformly bounded mass. Let \(R_\infty \) be a limit current of the last family. Without loss of generality, we can assume \(R_\infty = \lim _{k \rightarrow \infty } u_{1k} R_{2,k}\) and S is of bi-degree \((n-m,n-m)\). By standard arguments, we have \(R_\infty \le u_1 R_2\) (see [16, Proposition 3.2]). Thus, in order to have \(R_\infty = u_1 R_2\), we just need to check that
(both sides are finite because of the assumption we made at the beginning of the proof). Since \(\psi = 0\) on \(\partial U\) and \(u_{1k}=\psi \) on outside a compact of U, we have
Let \(u_{jk}^\epsilon , \psi ^\epsilon \) be standard regularizations of \(u_{jk}, \psi \), respectively. Since \(u_{jk}= \psi \) outside some compact of U, we have \(u_{jk}^\epsilon = \psi ^\epsilon \) outside some compact K of U, for \(\epsilon \) small enough and K independent of \(j,k,\epsilon \). Consequently, \(u_{jk}^\epsilon - \psi ^\epsilon \) is supported in \(K \Subset U\). Note that since \(\psi \) is smooth, \(\psi ^\epsilon \rightarrow \psi \) in \(\mathscr {C}^\infty \)- topology. By integration by parts and the fact that \(u_{jk} \ge u_j\) for \(j=1,2\), we have
by induction hypothesis for \((m-1)\) of (2.3) and the fact that \(\Vert \mathrm{dd}^c\psi _\epsilon - \mathrm{dd}^c\psi \Vert _{\mathscr {C}^0}= O(\epsilon )\). We now apply similar arguments to \(u_{3k}\) in place of \(u_{2k}\). Precisely, as above we have
Letting \(\epsilon \rightarrow 0\) and applying the induction hypothesis to the second term in the right-hand side of the last inequality (noticing again that \(\Vert \mathrm{dd}^c\psi _\epsilon - \mathrm{dd}^c\psi \Vert _{\mathscr {C}^0}= O(\epsilon )\)), we obtain
Put \(R'^\epsilon _{2,k}:= \mathrm{dd}^cu^\epsilon _{2k} \wedge \cdots \wedge \mathrm{dd}^cu^\epsilon _{mk}\). Repeating the above arguments for every \(u_{jk}\) (\(j \ge 2\)) and \(v,v_k\) gives
by (2.4) for \((m-1)\) and the usual convergence of Monge–Ampère operators. Letting \(k \rightarrow \infty \) in the last inequality and using (2.6) give (2.5). Hence, (2.3) for m follows.
It remains to prove (2.4) for m. Put \(R'_{2,k}:= \mathrm{dd}^cu_{2k} \wedge \cdots \wedge \mathrm{dd}^cu_{m k}\) and \(R'_2:= \mathrm{dd}^cu_{2} \wedge \cdots \wedge \mathrm{dd}^cu_{m}\). We check that \(Q_k\) is of uniformly bounded mass. Decompose
The second term converges to \(v \mathrm{dd}^c\psi \wedge R'_{2}\wedge S\) as \(k \rightarrow \infty \) by induction hypothesis for \((m-1)\). Denote by \(Q_{k,1}\) the first term. Let \(v^\epsilon _k\) be standard regularizations of \(v_k\). By integration by parts, we have
which converges to \(\int _U (u_{1k}- \psi ) R'_{2,k} \wedge \mathrm{dd}^cv_k\wedge S\) as \(\epsilon \rightarrow 0\) by (2.3) for m. Thus,
This combined with (2.3) for m again implies that \(\int _U Q_{k,1} \rightarrow \int _U (u_1 - \psi ) R'_2 \wedge \mathrm{dd}^cv \wedge S\) as \(k \rightarrow \infty \). The last limit is equal to \(\int _U v \mathrm{dd}^c(u_1 - \psi ) \wedge R'_2\) by integration by parts which can be performed thanks to (2.3) for m. Thus, we have proved that \(Q_k\) is of uniformly bounded mass and
as \(k \rightarrow \infty \). This combined with the fact that \(v R_1 \ge Q_\infty \) for every limit current \(Q_\infty \) of the family \((Q_k)_k\) gives the desired assertion (2.4) for m. This finishes the proof. \(\square \)
Remark 2.3
We can apply Theorem 2.2 to the case where S is a constant function and v is an arbitrary psh function. By the above proof, we can check the following observations:
(i) if S is a closed positive current on U, \(u_1, \ldots , u_m\) are psh functions on U which are locally integrable with respect to S such that \(u_j\) is locally bounded for every \(1 \le j \le m\) except possibly for one index, then the current \(\mathrm{dd}^cu_1 \wedge \cdots \wedge \mathrm{dd}^cu_m\wedge S\), which is defined inductively as usual, is symmetric with respect to \(u_1, \ldots , u_m\) and satisfies the convergence under decreasing sequences,
(ii) let \(u_0\) be another psh function locally integrable with respect to S such that \(u_0\) is locally bounded if there is an index \(1 \le j \le m\) so that \(u_j\) is not locally bounded. Then, \(u_0 \mathrm{dd}^cu_1 \wedge \cdots \wedge \mathrm{dd}^cu_m \wedge S\) is convergent under decreasing sequences and for every compact K in U, if we have \(0\le u_1,\ldots , u_m\le 1\), then
for some constant C independent of \(u_0, \ldots , u_m,S\).
The following result explains the reason for the use of Condition \((*)\).
Theorem 2.4
Let \((T_l)_l\) be a sequence of closed positive currents satisfying Condition \((*)\). Let u be a bounded psh function on U and \((u_k)_k\) a sequence of psh functions on U decreasing to u. Then for every constant \(\epsilon >0\) and every compact K in U, we have \(\text {cap}_{T_l}(\{|u_k -u| \ge \epsilon \}\cap K) \rightarrow 0\) as \(k \rightarrow \infty \) uniformly in l. In particular, for every constant \(\epsilon >0\), there exists an open subset \(U'\) of U such that \(\text {cap}_{T_l}(U',U)<\epsilon \) for every l and the restriction of u to \(U \backslash U'\) is continuous.
Consider the case where \(T_l= T\) for every l. Then, the above theorem gives a quasi-continuity with respect to \(\text {cap}_{T}\) for bounded psh function which is stronger than the usual one for general psh functions with respect to \(\text {cap}\) (see [1]). We refer to Theorem 2.4 as a (uniform) strong quasi-continuity of bounded psh functions.
Proof
We follow ideas presented in [18, Proposition 1.12]. Let \(K \Subset U\). Let \(T_l\) be of bi-dimension (m, m). We will prove that
uniformly in psh functions \(0 \le v_1, \ldots , v_m \le 1\) and in l. The desired assertion concerning the uniform convergence in \(\text {cap}_{T_l}\) is a direct consequence of (2.7).
By Hartog’s lemma and the boundedness of u, we obtain that \(u_k\) is uniformly bounded in k in compact subsets of U. Since the problem is local and \(u_k,u, v_1, \ldots , v_m\) are uniformly bounded on U, we can assume that \(U\Subset {\mathbb {C}}^n\), \(u_k,u, v_1, \ldots , v_m\) are defined on an open neighborhood of \({\overline{U}}\) and there exist a smooth psh function \(\psi \) defined on an open neighborhood of \({\overline{U}}\) and an open neighborhood W of \(\partial U\) such that \(K \subset U \backslash W\) and \(u_k=u=v_j=\psi \) on W for every j, k. Let
Observe \(u_k -u\) is of compact support in some open set \(U_1 \Subset U\) containing K. Hence, by integration by parts, we get
which is \( \lesssim \bigg (\int _{U_1} d(u_k - u) \wedge \mathrm{d}^c(u_k -u) \wedge T'_l\bigg )^{1/2}\) by the Chern–Levine–Nirenberg inequality. Denote by I the integral in the last quantity. We have
Applying similar arguments to \(v_2,\ldots , v_m\) consecutively and the right-hand side of the last inequality, we obtain that
where C is independent of k and l (note that the mass of \(T_l\) on compact subsets of U is bounded uniformly in l). Let
By (2.8), in order to obtain (2.7), it suffices to prove that \(H_{k,l}\) converges to 0 as \(k \rightarrow \infty \) uniformly in l. Suppose that this is not the case. This means that there exists a constant \(\epsilon >0\), \((k_s)_s\rightarrow \infty \) and \((l_s)_s\rightarrow \infty \) such that \(H_{k_s, l_s} \ge \epsilon \) for every s. However, by Condition \((*)\), we get \((u_{k_s} -u) (\mathrm{dd}^cu)^{m} \wedge T_{l_s} \rightarrow 0\) as \(s \rightarrow \infty \). This is contradiction. Hence, (2.7) follows.
We prove the second desired assertion, let \(K \Subset U\) and \((u_k)_k\) a sequence of smooth psh functions defined on an open neighborhood of K decreasing to u. Let \(\epsilon >0\) be a constant. Since \(u_k \rightarrow u\) in \(\text {cap}_{T_l}\) as \(k \rightarrow \infty \) uniformly in l, there is a sequence \((j_k)_k\) converging to \(\infty \) for which
for every \(k,l \in {\mathbb {N}}^*\). Consequently, for \(K_\epsilon := K \backslash \cup _{k=1}^\infty \{ u_{j_k} > u+ 1/k \}\), we have that \(\text {cap}_{T_l}(K\backslash K_\epsilon ,U) \le \epsilon \) and \(u_{j_k}\) is convergent uniformly on \(K_\epsilon \). Hence, u is continuous on \(K_\epsilon \).
Let \((U_s)_s\) be an increasing exhaustive sequence of relatively compact open subsets of U and \(K_s:= {\overline{U}}_s \backslash U_{s-1}\) for \(s \ge 1\), where \(U_0:= \varnothing \). Observe that \(K_l\) is compact, \(U= \cup _{s=1}^\infty K_s\) and
for every \(s \ge 1\). By the previous paragraph, there exists a compact subset \(K'_s\) of \(K_s\) such that \(\text {cap}_{T_l}(K_s \backslash K'_s,U) \le \epsilon 2^{-s}\) and u is continuous on \(K'_s\). Observe that \(K':=\cup _{s=1}^\infty K'_s\) is closed in U and u is continuous on \(K'\) because of (2.9). We also have \(U \backslash K' \subset \cup _{s=1}^\infty (K_s \backslash K'_s)\). Hence, \(\text {cap}_{T_l}(U\backslash K', U)\le \epsilon \) for every l. The proof is finished. \(\square \)
As one can expect, the above quasi-continuity of bounded psh functions allows us to treat, to certain extent, these functions as continuous functions with respect to closed positive currents.
Corollary 2.5
Let \(R_k:= \mathrm{dd}^cv_{1k} \wedge \cdots \wedge \mathrm{dd}^cv_{m k} \wedge T_k\) and \(R:= \mathrm{dd}^cv_{1} \wedge \cdots \wedge \mathrm{dd}^cv_{m} \wedge T\), where \(v_{j k}, v_j\) are uniformly bounded psh functions on U and \(T_k,T\) closed positive currents of bi-degree (p, p). Let u be a bounded psh function on U and \(\chi \) a continuous function on \({\mathbb {R}}\). Assume that \(R_k \rightarrow R\) as \( k\rightarrow \infty \) on U and \((T_k)_k\) satisfies Condition \((*)\). Then, we have
as \(k \rightarrow \infty \). In particular, the last convergence holds when \(T_k= T\) for every k or \(T_k = \mathrm{dd}^cw_k\wedge S\), \(T= \mathrm{dd}^cw\wedge S\), where S is a closed positive current, w is a psh function locally integrable with respect to S , and \(w_k\) is a psh function converging to w in \(L^1_{loc}\) as \(k \rightarrow \infty \) so that \(w_k \ge w\) for every k.
Proof
The problem is local. Hence, we can assume U is relatively compact in \({\mathbb {C}}^n\). Since u is bounded, using Theorem 2.4, we have that u is uniformly quasi-continuous with respect to the family \(\text {cap}_{T_k}\) with \(k \in {\mathbb {N}}\). This means that given \(\epsilon >0\), we can find an open subset \(U' \) of U such that \(\text {cap}_{T_k}(U',U) < \epsilon \) and \(u|_{U\backslash U'}\) is continuous. Let \({\tilde{u}}\) be a bounded continuous function on U extending \(u|_{U\backslash U'}\) (see [20, Theorem 20.4]). We have \(\chi ({\tilde{u}}) R_k \rightarrow \chi ({\tilde{u}}) R\) because \(\chi , {\tilde{u}}\) are continuous. Moreover,
(we used here the boundedness of U) and a similar estimate also holds for \(\big (\chi ({\tilde{u}})- \chi (u)\big )R\). The desired assertion then follows. This finishes the proof. \(\square \)
The following result generalizes well-known convergence properties of Monge–Ampère operators in [1].
Theorem 2.6
Let \(U \subset {\mathbb {C}}^n\) be an open set. Let \((T_k)_k\) be a sequence of closed positive currents satisfying Condition \((*)\) so that \(T_k\) converges to a closed positive current T on U as \(k \rightarrow \infty \). Let \(u_j\) be a locally bounded psh function on U for \(1 \le j \le m\). Let \((u_{jk})_{k \in {\mathbb {N}}}\) be a sequence of locally bounded psh functions converging to \(u_j\) in \(L^1_{loc}\) as \(k \rightarrow \infty \). Then, the convergence
as \(k \rightarrow \infty \) holds provided that one of the following conditions is fulfilled:
-
(i)
\(u_{jk}(x) \nearrow u_j(x)\) for every \(x\in U\) as \(k \rightarrow \infty \),
-
(ii)
\(u_{jk}(x) \nearrow u_j(x)\) for almost everywhere \(x \in U\) (with respect to the Lebesgue measure) and T has no mass on pluripolar sets,
-
(iii)
\(u_{jk} \ge u_j\) for every j, k.
Proof
Given that we already have a uniform strong quasi-continuity for bounded psh functions, the desired result can be deduced without difficulty from proofs of classical results on the convergence of Monge–Ampère operators, for example, see [18].
We present here a proof of Theorem 2.6 for the readers’ convenience. First of all, observe that if \(u_{jk} \nearrow u_j\) almost everywhere then, we have \(u_{jk} \le u_{j (k+1)} \le u_j\) pointwise on U and the set \(\{x \in U: u_j(x) \not = \lim _{k\rightarrow \infty } u_{jk}(x) \}\) is pluripolar. By the localization principle ([18, Page 7]), we can assume that \(u_{jk}, u_j\) are all equal to some smooth psh function \(\psi \) outside some set \(K \Subset U\) on U. We prove (i), (ii) simultaneously. Let
We prove by induction in j that
k and for every \(2 \le j \le m+1\) (by convention we put \(S_{(m+1)k}:= T_k\) and \(S_{m+1}:= T\)). The claim is true for \(j=m+1\). Suppose that it holds for \((j+1)\). We need to prove it for j. Let \(R_{j \infty }\) be a limit current of \(u_{(j-1)k} S_{jk}\) as \(k \rightarrow \infty \). By induction hypothesis (2.11) for \((j+1)\) instead of j, \(S_{j k} \rightarrow S_{j}\) as \(k \rightarrow \infty \). This combined with the fact that the sequence \((u_{jk})_k\) converges in \(L^1_{loc}\) to \(u_j\) gives
(one can see [16, Proposition 3.2]). On the other hand, since \((u_{jk})_k\) is increasing, using Corollary 2.5, we obtain
for every \(s \in {\mathbb {N}}\). Letting \(s \rightarrow \infty \) in the last inequality gives
Recall that the set of \(x \in U\) with \(u_{j-1}(x)> \lim _{s \rightarrow \infty }u_{(j-1)s}(x) \) is empty in the setting of (i) and is a pluripolar set in the setting of (ii). Hence, (2.11) follows from Lemma 2.1, (2.13) and (2.12). We have proved (i) and (ii). We prove (iii) by similar induction. The proof is finished. \(\square \)
Remark 2.7
By the above proof and Lemma 2.1, Property (ii) of Theorem 2.6 still holds if instead of requiring T has no mass on pluripolar sets, we assume the following two conditions:
(i) T has no mass on \(A_j:= \{x \in U: u_j(x) \not = \lim _{k\rightarrow \infty } u_{jk}(x) \}\) for every \(1 \le j \le m\) and,
(ii) the set \(A_j\) is locally complete pluripolar for every j.
Just by replacing the usual quasi-continuity of psh functions by the stronger one given in Theorem 2.4 for bounded psh functions, we immediately obtain results similar to those in [2]. We state here results we will use later.
Lemma 2.8
(similar to [2, Lemma 4.1]) Let U be an open subset in \({\mathbb {C}}^n\). Let T be a closed positive current on U and \(u_j, u_{jk}, u'_j, u'_{jk}\) bounded psh functions on U for \(k \in {\mathbb {N}}\) and \(1 \le j \le m\), where \(m \in {\mathbb {N}}\). Let \(q \in {\mathbb {N}}^*\) and \(v_j, v'_j\) bounded psh functions on U for \(1 \le j \le q\). Put \(W:= \cap _{j=1}^q \{v_j > v'_j\}\). Assume that
and
as \(k \rightarrow \infty \) and
for every k. Then we have \(\mathbf{1} _{W} R=\mathbf{1} _{W} R'.\)
Proof
The problem is clear if W is open, for example, when \(v_j\) is continuous for \(1 \le j \le q\). In the general case, we will use the strong quasi-continuity to modify \(v_j\). Since the problem is local, we can assume that U is bounded. Let \(\epsilon >0\) be a constant. By Theorem 2.4, we can find bounded continuous functions \({\tilde{v}}_j\) on U such that \(\text {cap}_T(\{{\tilde{v}}_j\not = v_j\},U)< \epsilon \). Put \({\tilde{W}}:= \cap _{j=1}^q \{{\tilde{v}}_j > v'_j\}\) which is an open set. The choice of \({\tilde{v}}_j\) combined with the definition of \(\text {cap}_T\) yields that
We also have similar estimates for \(R', R'_k\). By this and (2.14), we get \( \Vert \mathbf{1} _{{\tilde{W}}} R_k-\mathbf{1} _{{\tilde{W}}} R'_k \Vert _U \le 2 \epsilon \). This combined with the fact that \({\tilde{W}}\) is open yields that \(\Vert \mathbf{1} _{{\tilde{W}}} R-\mathbf{1} _{{\tilde{W}}} R' \Vert _U \le 2 \epsilon \). Thus, \(\Vert \mathbf{1} _{W} R-\mathbf{1} _{W} R' \Vert _U \le 4 \epsilon \) for every \(\epsilon \). The desired equality follows. This finishes the proof. \(\square \)
Theorem 2.9
Let U be an open subset in \({\mathbb {C}}^n\). Let T be a closed positive current on U and \(u_{j}, u'_j\) bounded psh functions on U for \(1 \le j \le m\), where \(m \in {\mathbb {N}}\). Let \(v_j,v'_j\) be bounded psh functions on U for \(1 \le j \le q\). Assume that \(u_{j}= u'_{j}\) on \(W:=\cap _{j=1}^q\{v_j> v'_j\}\) for \(1 \le j \le m\). Then we have
Proof
We give here a complete proof for the readers’ convenience. Let \(\epsilon >0\) be a constant. Put \(u''_j:= \max \{u_j, u'_j-\epsilon \}\) and \({\tilde{W}}:= \cap _{j=1}^m \{u_j > u'_j- \epsilon \}\). By hypothesis, \(W \subset {\tilde{W}}\). We will prove that
Since the problem is local, we can assume there is a sequence of uniformly bounded smooth psh functions \((u_{j k})_k\) decreasing to \(u_j\) for \(1 \le j \le m\). Since \({\tilde{W}}_{k}:= \{u_{jk} >u'_j- \epsilon \}\) is open, we have
This together with the inclusion \({\tilde{W}} \subset {\tilde{W}}_k\) gives
Using this and Lemma 2.8, we obtain (2.16) by considering \(k \rightarrow \infty \). In particular, we get
Letting \(\epsilon \rightarrow 0\) and using Lemma 2.8 again gives
The last equality still holds if we replace \(u_j\) in the left-hand side by \(u'_j\) by using similar arguments. So the desired equality follows. The proof is finished. \(\square \)
Remark 2.10
Recall that a quasi-psh function u on U is, by definition, locally the sum of a psh function and a smooth one. We can check that results presented above have their analogues for quasi-psh functions.
3 Relative non-pluripolar product
Let X be a complex manifold of dimension n and \(T, T_1, \ldots , T_m\) closed positive currents on X such that \(T_j\) is of bi-degree (1, 1) for \(1 \le j \le m\). Let U be a local chart of X such that \(T_j= \mathrm{dd}^cu_j\) on U for \(1 \le j \le m\), where \(u_1, \ldots , u_m\) are psh functions on U. Let \(k \in {\mathbb {N}}\) and \(u_{jk}:= \max \{u_j, -k\}\) which is a locally bounded psh function. Put \(R_k:= \mathrm{dd}^cu_{1k} \wedge \cdots \wedge \mathrm{dd}^cu_{mk} \wedge T\). By Theorem 2.9 and the fact that \(\{u_j> -k\}= \{u_{jk} > -k\}\), we have
for every \(l \ge k\). As in the case of the usual non-pluripolar products, we have the following basic observation.
Lemma 3.1
Assume that we have
for every compact K of U. Then the limit current
is well defined and for every Borel form \(\Phi \) with bounded coefficients on U such that \(\mathrm{Supp}\Phi \Subset U\), we have
Consequently, there holds
Proof
By (3.1), we have
This combined with (3.2) tells us that the mass on a fixed compact of U of the current
converging to 0 as \(l\ge k\rightarrow \infty \). We deduce that \(\lim _{k\rightarrow \infty }{} \mathbf{1} _{\cap _{j=1}^m \{u_j > -k\}} R_k\) exists and is denoted by R.
Since \(R, R_k\) are positive, for every continuous form \(\Phi \) of compact support in U, we have \(\langle R, \Phi \rangle = \lim _{k \rightarrow \infty } \langle R_k, \Phi \rangle \). Let \(\Phi \) be a Borel form on U such that its coefficients are bounded on U and \(\mathrm{Supp}\Phi \Subset U\). Let K be a compact of U containing \(\mathrm{Supp}\Phi \) and \(U_1 \supset K\) a relatively compact open subset of U. Let \(\epsilon >0\) be a constant. Let \(k_0\) be a positive integer such that
for every \(l\ge k \ge k_0\). By Lusin’s theorem, there exists a continuous form \(\Phi '\) compactly supported on \(U_1\) such that
This combined with the fact that \(|\langle |\mathbf{1} _{\cap _{j=1}^m \{u_j > -k\}} R_k, \Phi '\rangle \rightarrow \langle R, \Phi '\rangle \) gives
Letting \(\epsilon \rightarrow 0\) gives (3.4). For the other equalities, one just needs to apply (3.4) to suitable \(\Phi \). This finishes the proof. \(\square \)
Lemma 3.2
Let \(T_j= \mathrm{dd}^c{\tilde{u}}_j+ \theta _j\) for \(1 \le j \le m\), where \(\theta _j\) is a smooth (1, 1)-form and \({\tilde{u}}_j\) is \(\theta _j\)-psh on X. Let
Then the following two properties hold:
-
(i)
(3.2) holds for every small enough local chart U if and only if we have that for every compact K of X,
$$\begin{aligned} \sup _{k \in {\mathbb {N}}} \Vert \mathbf{1} _{\cap _{j=1}^m \{{\tilde{u}}_j >-k\}} {\tilde{R}}_k\Vert _K < \infty . \end{aligned}$$(3.7)In this case, if \({\tilde{R}}:= \lim _{k \rightarrow \infty } \mathbf{1} _{\cap _{j=1}^m \{{\tilde{u}}_j >-k\}}{\tilde{R}}_k\), then \({\tilde{R}}=R\) on U,
-
(ii)
\(\mathbf{1} _{\cap _{j=1}^m \{{\tilde{u}}_j >-k\}} {\tilde{R}}_k\) is a positive current.
Proof
Firstly observe that \({\tilde{R}}_k\) is a current of order 0 and of bounded mass on compact subsets of X. Let
Assume now (3.7). This means \(\Vert \mathbf{1} _{{\tilde{B}}_k} {\tilde{R}}_k\Vert _K\) is uniformly bounded for every compact K. By Remark 2.10, we have \(\mathbf{1} _{{\tilde{B}}_k} {\tilde{R}}_k= \mathbf{1} _{{\tilde{B}}_k}{\tilde{R}}_l\) for every \(l\ge k\). Decompose
which is a disjoint union. Hence, we get
where the masses are measured on some compact K in X. We deduce that the condition \(\Vert \mathbf{1} _{{\tilde{B}}_k} {\tilde{R}}_k\Vert _K\) is uniformly bounded is equivalent to that
as \(l \ge k \rightarrow \infty \). Hence, \(\mathbf{1} _{{\tilde{B}}_k} {\tilde{R}}_k\) converges to a current \({\tilde{R}}\) and moreover we have
for every Borel bounded form \(\Phi \) of compact support in X as in the proof of Lemma 3.1. Consequently, we get
Let \(U, u_j, u_{jk}, R, R_k\) be as above. We will show that (3.2) is satisfied and \({\tilde{R}}= R\) on U. Let
Observe that \(u_j={\tilde{u}}_j+ \tau _j\) for some smooth function \(\tau _j\) on U with \(\mathrm{dd}^c\tau _j = \theta _j\). By shrinking U, we can assume that \(\tau _j\) is bounded on U and let \(c_0\) be an integer greater than \(\sum _{j=1}^m \Vert \tau _j\Vert _{L^\infty }\). We have
which is equal to \(\max \{ {\tilde{u}}_j+ \tau _j, -k \}= u_{jk}\) on the set \(\{{\tilde{u}}_j > -k + c_0\}\). It follows that
This together with the inclusions \(B_{k-2 c_0} \subset {\tilde{B}}_{k-c_0} \subset B_k\) give
Hence (3.2) follows. We also deduce from this and (3.9) that \({\tilde{R}}=R\). Conversely, if (3.2) holds for every U, then, by (3.11), the claim (3.7) holds. Hence, (i) follows. By (3.10), we have
Thus (ii) follows. The proof is finished. \(\square \)
Lemma 3.2 applied to U in place of X implies that the condition (3.2) is independent of the choice of \(u_j\) and so is the limit R above. As a result, if (3.2) holds for every small enough local chart U as above, then we obtain a positive current R globally defined on X given locally by (3.3). This current is equal to \({\tilde{R}}\) by Lemma 3.2 again.
Definition 3.3
We say that the non-pluripolar product relative to T of \(T_1, \ldots , T_m\) is well defined if (3.2) holds for every small enough local chart U of X, or equivalently, (3.7) holds. In this case, the non-pluripolar product relative to T (or the T-relative non-pluripolar product) of \(T_1, \ldots , T_m\), which is denoted by \(\langle T_{1} \wedge \cdots \wedge T_{m} {\dot{\wedge }} T \rangle \), is defined to be the current \({\tilde{R}}\) in Lemma 3.2.
Note that \(\langle T_{1} \wedge \cdots \wedge T_{m} {\dot{\wedge }} T \rangle \) is a current of bi-degree \((p+m,p+m)\) if T is of bi-degree (p, p). If \(T_1, \ldots , T_m\) are of locally bounded potentials, then \(\langle T_{1} \wedge \cdots \wedge T_{m} {\dot{\wedge }} T \rangle \) is equal to the classical intersection of \(T_1, \ldots , T_m\) and T, see Proposition 3.6 for more information.
When T is the current of integration along X, we write \(\langle T_1 \wedge \cdots \wedge T_m \rangle \) for the non-pluripolar product relative to T of \(T_1, \ldots , T_m\). One can see that this is exactly the usual non-pluripolar product of \(T_1, \ldots , T_m\) defined in [2, 4, 17]. We note that in general the current \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) can have mass on pluripolar sets in \(\cap _{j=1}^m \{u_j >-\infty \}\), see, however, Property (iii) of Proposition 3.5 below. Arguing as in the proof of [4, Proposition 1.6], we obtain the following result.
Lemma 3.4
Assume that X is a compact Kähler manifold. Then, \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is well defined.
For a closed positive (1, 1)-current R, recall that the polar locus \(I_R\) of R is the set of points where potentials of R are equal to \(-\infty \). Note that by Lemma 3.1, the current \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle \) has no mass on \(\cup _{j=1}^m I_{T_j}\). We collect here some more basic properties of relative non-pluripolar products.
Proposition 3.5
-
(i)
The product \( \langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle \) is symmetric with respect to \(T_1, \ldots , T_m\).
-
(ii)
Given a positive real number \(\lambda \), we have \(\langle (\lambda T_1) \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle = \lambda \langle T_1 \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle \).
-
(iii)
Given a complete pluripolar set A such that T has no mass on A, then \(\langle T_1 \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle \) also has no mass on A.
-
(iv)
Let \(T'_1\) be a closed positive (1, 1)-current on X and \(T_j,T\) as above. Assume that \(\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T\rangle \), \(\langle T'_1 \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T\rangle \) are well defined. Then, \(\langle (T_1+T'_1) \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T\rangle \) is also well defined and satisfies
$$\begin{aligned} \left \langle (T_1+T'_1) \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T \right \rangle \le \left \langle T_1 \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }}T \right \rangle + \left \langle T'_1 \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T \right \rangle . \end{aligned}$$(3.12)The equality occurs if T has no mass on \(I_{T_1} \cup I_{T'_1}\).
-
(v)
Let \(1 \le l \le m\) be an integer. Let \(T''_j\) be a closed positive (1, 1)-current on X and \(T_j,T\) as above for \(1 \le j \le l\). Assume that \(T''_j \ge T_j\) for every \(1 \le j \le l\) and T has no mass on \(\cup _{j=1}^l I_{T''_j-T_j}\). Then, we have
$$\begin{aligned} \langle \bigwedge _{j=1}^l T''_j \wedge \bigwedge _{j=l+1}^m T_j {\dot{\wedge }} T \rangle \ge \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \end{aligned}$$provided that the left-hand side is well defined.
-
(vi)
Let \(1 \le l \le m\) be an integer. Assume \(R:= \langle \bigwedge _{j=l+1}^m T_j {\dot{\wedge }} T \rangle \) and \(\langle \bigwedge _{j=1}^l T_j {\dot{\wedge }} R \rangle \) are well defined. Then, we have \(\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle = \langle \bigwedge _{j=1}^l T_j {\dot{\wedge }} R \rangle \).
-
(vii)
Let A be a complete pluripolar set. Assume that \(\langle T_1 \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle \) is well defined. Then, we have
$$\begin{aligned} \mathbf{1} _{X \backslash A}\langle T_1 \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }} T\rangle = \big \langle T_1 \wedge T_2 \wedge \cdots \wedge T_m {\dot{\wedge }}(\mathbf{1} _{X \backslash A} T)\big \rangle . \end{aligned}$$In particular, the equality
$$\begin{aligned} \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle = \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T' \rangle \end{aligned}$$holds, where \(T':= \mathbf{1} _{X \backslash \cup _{j=1}^m I_{T_j}} T\).
Proof
Properties (i), (ii) are clear from the definition and the proof of Lemma 3.1. We now check (iii). Let \(R:= \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T\rangle \). Since T has no mass on the complete pluripolar set A, using Lemma 2.1 gives \(\mathbf{1} _{A}\bigwedge _{j=1}^m \mathrm{dd}^cu_{jk} \wedge T=0\) for every k. Using this and Lemma 3.1, we deduce that
Hence \(R=0\) on A and (iii) follows.
We prove (iv). We work on a small local chart U. Write \(T'_j= \mathrm{dd}^cu'_j\), \(u'_{jk}:= \max \{u'_j, -k\}\). Recall \(T_j= \mathrm{dd}^cu_j\). We can assume \(u_j, u'_j \le 0\). Let \(R':= \langle T'_1 \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T\rangle \) and \(R'':= \langle (T_1+T'_1) \wedge \bigwedge _{j=2}^m T_j {\dot{\wedge }} T\rangle \). We have
on \(\{u_1 + u'_1 >-k\}\) because the last set is contained in \(\{u_1> -k\} \cap \{u'_1 > -k \}\). This combined with Lemma 3.1 yields
where \(B''_k: =\{u_1+ u'_1> -k\} \cap \cap _{j=2}^m \{u_j > -k\}\). Letting \(k \rightarrow \infty \) in the last equality gives (3.12). Observe that
as \(k \rightarrow \infty \). The last limit is equal to \(\mathbf{1} _{I_{T'_1}} R\) because R has no mass on \(\cup _{j=1}^m I_{T_j}\). Moreover since \(I_{T_1'}\) is complete pluripolar and T has no mass on \(I_{T'_1}\), by (iii), we have that \(\mathbf{1} _{I_{T'_1}} R =0\). Consequently, \(\mathbf{1} _{U \backslash B''_k } R \rightarrow 0\) as \(k \rightarrow \infty \) and a similar property of \(R'\) also holds. Thus, we obtain the equality in (3.12) if T has no mass on \(I_{T_1} \cup I_{T'_1}\). To get (v), we just need to decompose \(T''_j= T_j+ T'_j\) for some closed positive current \(T'_j\) and use similar arguments as in the proof of (iv).
We prove (vi). We can assume \(u_j \le 0\) for every \(1 \le j \le m\). Let
Observe that \(0 \le \psi _k \le 1\) and
Note that R has no mass on \(I_{T_j}\) for \(l+1 \le j \le m\). This combined with (iii) yields that \(\langle \bigwedge _{j=1}^l T_j {\dot{\wedge }} R \rangle \) gives no mass on \(I_{T_j}\) for \(1 \le j \le m\). Using this and the fact that \(\psi _k \nearrow \mathbf{1} _{X \backslash \cup _{j=1}^m I_{T_j}}\) yields
Now since \(\psi _k R = \psi _k \bigwedge _{j=l+1}^m \mathrm{dd}^cu_{jk} \wedge T\) (Lemma 3.1), we get
To see why the last equality is true, notice that it is clear if \(u_{jk}\)’s are smooth and in general, we can use sequences of smooth psh functions decreasing to \(u_{jk}\) for \(1 \le j \le l\) and the convergences of Monge–Ampère operators to obtain (3.14). Combining (3.14) with (3.13) gives the desired assertion.
It remains to prove (vii). Let \(\psi _k\) be as above. Let \(A= \{\varphi = -\infty \}\), where \(\varphi \) is a negative psh function. Define \(\psi '_k:=k^{-1} \max \big \{ \varphi + \sum _{j=1}^m u_j, -k \big \}+ 1.\) Since \(0 \le \psi '_k \le \psi _k\), we get \(\{\psi _k= 0\} \subset \{\psi '_k=0\}\). It follows that
because \(T = \mathbf{1} _{X \backslash A} T\) on \(\{\psi '_k \not = 0\}\) and the convergence of Monge–Ampère operators. Letting \(k \rightarrow \infty \) gives the desired assertion. The proof is finished. \(\square \)
The following result clarifies the relationship between the relative non-pluripolar product and some other known notions of intersection.
Proposition 3.6
-
(i)
Let \(U, u_j, u_{jk}, R_k\) be as above. Assume that \(u_j\) is locally integrable with respect to \(T_{j+1} \wedge \cdots \wedge T_m \wedge T\) for \(1\le j \le m\) and for every bounded psh function v, the current \(v R_k\) converges to \(v \, T_1 \wedge \cdots \wedge T_m \wedge T\) on U as \(k \rightarrow \infty \). Then, the current \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is well defined on U and
$$\begin{aligned} \langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle = \mathbf{1} _{X \backslash \cup _{j=1}^m I_{T_j}} T_1 \wedge \cdots \wedge T_m \wedge T. \end{aligned}$$(3.15)In particular, if \(u_1, \ldots , u_{m-1}\) are locally bounded and \(u_m\) is locally integrable with respect to T, then we have
$$\begin{aligned} \langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle = \mathbf{1} _{X\backslash I_{T_m}} T_1 \wedge \cdots \wedge T_m \wedge T. \end{aligned}$$(3.16) -
(ii)
If T is of bi-degree (1, 1), then we have
$$\begin{aligned} \langle T_1 \wedge \cdots \wedge T_m \wedge T \rangle = \mathbf{1} _{X \backslash I_{T}}\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle , \end{aligned}$$(3.17)where the left-hand side is the usual non-pluripolar product of \(T_1, \ldots , T_m,T\).
We note that in the above (i), the current \(T_j \wedge \cdots \wedge T_m \wedge T\) \((1 \le j \le m)\) is defined inductively as in the classical case (see [1]). The assumption of (i) of Proposition 3.6 is satisfied in well-known classical contexts; we refer to [16] and references therein for details. We notice also that by (vii) of Proposition 3.5, the right-hand side of (3.17) is equal to \(\big \langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} (\mathbf{1} _{X \backslash I_{T}}T) \big \rangle \).
Proof
We prove (i). The question is local. Hence, we can assume \(u_j \le 0\) for every j. Let \(\psi _k:= k^{-1} \max \{\sum _{j=1}^m u_j, -k \}+1\). By hypothesis, we get
as \(r \rightarrow \infty \). Since \(\psi _k=0\) on \(\{u_j \le -k\}\) and \(u_{jr}=u_{jk}\) on \(\{u_j > -k \}\) for \(r \ge k\), using Theorem 2.9, we infer that \(\psi _k R_r =\psi _k R_k.\) Consequently, the mass \(\psi _k R_k\) on compact subsets of U is bounded uniformly in k. Hence, \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is well defined on U. The above arguments also show that
which is equal to \(\psi _k \langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle .\) Letting \(r \rightarrow \infty \) in the last equality and using (3.18) give
Letting \(k \rightarrow \infty \) gives (3.15). To obtain (3.16), we just need to combine (3.15) with Theorem 2.2.
It remain to check (ii). We work locally. Let u be a local potential of T and \(u \le 0\). Let \(u_k:= \max \{u,-k\}\) and \(\psi '_k:= k^{-1} \max \{u+\sum _{j=1}^m u_j, -k \}+1\). Note that
Since \(\{\psi _k =0\} \subset \{\psi '_k =0\}\) (\(\psi _k \ge \psi '_k\ge 0\)), we have
(one can obtain the last equality as a consequence of (3.16) applied to the case where T is the current of integration along X or alternatively we can use Theorem 2.2 directly). Letting \(k \rightarrow \infty \) in the last equality gives (3.17). This finishes the proof. \(\square \)
As in the case of the usual non-pluripolar products, the relative non-pluripolar products, if well defined, are closed positive currents as showed by the following result.
Theorem 3.7
Assume that \(\langle T_1 \wedge \cdots \wedge T_m {\dot{\wedge }} T \rangle \) is well defined. Then, \(\langle T_{1} \wedge \cdots \wedge T_{m} {\dot{\wedge }} T \rangle \) is closed.
Proof
The proof is based on ideas from [4, 21]. We work on a local chart U as above. By shrinking U and subtracting from \(u_j\) a suitable constant, we can assume that \(u_j \le 0\). Let
Observe that \(\psi _k= 0\) on \(\cup _{j=1}^m \{u_j \le -k\}\) and \(0 \le \psi _k \le 1\) increases to \(\mathbf{1} _{\cap _{j=1}^m \{u_j > -\infty \}}\). Let \(g: {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a nonnegative smooth function such that \(g(0)=g'(0)=g'(1)=0\) and \(g(1)=1\). Let \(R_k, R\) be as above. Since \(g(1)=1\) and \(g(0)=0\), we get \(g(\psi _k)R \rightarrow R\) as \(k \rightarrow \infty \) (recall R has no mass on \(\cup _{j=1}^m \{u_j = -\infty \}\)). Thus, the desired assertion is equivalent to proving that
Since \(g(\psi _k)=g(0)=0\) on \(\cup _{j=1}^m \{u_j \le -k\}\), we have
(see [4, Lemma 1.9] or Corollary 2.5 for the second equality). Let \(U_1 \Subset U_2\) be relatively compact open subsets of U. By the Cauchy–Schwarz inequality, we have
Using \( d \psi _k \wedge \mathrm{d}^c\psi _k= \mathrm{dd}^c\psi _k^2 - \psi _k \mathrm{dd}^c\psi _k\) and the Chern–Levine–Nirenberg inequality, one get
by (3.2). On the other hand, using the equality \(g'(\psi _k)= g'(0)=0\) on \(\{u_j \le -k\}\), we obtain
converging to 0 as \(k \rightarrow \infty \) because \(g'(\psi _k) \rightarrow g'(1)= 0\) on \(\cap _{j=1}^m\{u_j >-\infty \}\). Thus, (3.19) follows. This finishes the proof. \(\square \)
Remark 3.8
Let V be a smooth submanifold of X and T the current of integration along V. If \(V \subset \cup _{j=1}^m I_{T_j}\), then the non-pluripolar product relative to T of \(T_1, \ldots , T_m\) is zero. Consider now the case where \(V \not \subset \cup _{j=1}^m I_{T_j}\). In this case, we can define a current \(T'_j\) which can be viewed as the intersection of \(T_j\) and T as follows. Let \(u_1, \ldots , u_m\) be local potentials of \(T_1, \ldots , T_m\), respectively. Let \(T'_j:= \mathrm{dd}^c(u_j|_V)\) for \(1 \le j \le m\). One can see that \(T'_j\) is independent of the choice of \(u_j\); hence, \(T'_j\) is a well-defined closed positive (1, 1)-current on V. Let \(\iota : V \hookrightarrow X\) be the natural inclusion. We can check that \(\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \) is equal to the pushforward by \(\iota \) of \(\langle \bigwedge _{j=1}^m T'_j \rangle \).
4 Monotonicity
In this section, we present a monotonicity property for relative non-pluripolar products. We begin with the following simple lemma.
Lemma 4.1
Let U be an open subset of \({\mathbb {C}}^n\). Let \((u^l_{j})_l\) be a sequence of psh functions on U and \(u_j\) a psh function on U for \(1 \le j \le m\). Let \((T_l)_l\) be a sequence of closed positive currents satisfying Condition \((*)\) and \(T_l\) converges to a closed positive current T on U as \(l \rightarrow \infty \). Assume that one of the following conditions is satisfied:
-
(i)
\(u_{j}^l \ge u_j\) and \(u_{j}^l\) converges to \(u_j\) in \(L^1_{loc}\) as \(l \rightarrow \infty \).
-
(ii)
\(u_{j}^l\) increases to \(u_j\) as \(l \rightarrow \infty \) almost everywhere and T has no mass on pluripolar sets.
Then, for every smooth weakly positive form \(\Phi \) with compact support in U, we have
When T is the current of integration along X, a related statement in the compact setting was given in [9, Theorem 2.3].
Proof
Let \(\Phi \) be a smooth weakly positive form with compact support in U. Assume (i) holds. Let \(u^l_{j k}:= \max \{u_{j}^l, -k\}\) which converges to \(u_{j k}:= \max \{u_j, -k\}\) in \(L^1_{loc}\) as \(l \rightarrow \infty \) and \(u^l_{jk} \ge u_{jk}\). Put
Similarly, we define \(R, R_k\) by using the formula of \(R^l,R^l_k\), respectively, with \(u_j^l, u_{jk}^l\) replaced by \(u_j, u_{jk}\). Since \(u_j^l \ge u_j\), we get \(\{u_j>- k\} \subset \{u_j^l >-k\}\). It follows that
By Theorem 2.6, we get \(R^l_k \rightarrow R_k\) as \(l \rightarrow \infty \). Using this together with the fact that when k is fixed, \(u_{jk}^l\) is uniformly bounded in l, we see that the strong uniform quasi-continuity for \(u_{jk}\) with respect to \((T_l)_l\) (see Theorem 2.4, and also [2, Corollary 3.3]) implies
This combined with (4.1) yields
for every k. Letting \(k \rightarrow \infty \) in the last inequality and noticing that R has no mass on \(\cup _{j=1}^m \{u_j= -\infty \}\) give the desired assertion.
Now assume (ii). Note that
for every l, j. Thus, \(\{u^l_j>-k\} \subset \{u^{l+1}_j>-k\} \subset \{u_j >-k\}\) and \({\tilde{u}}_j:= \lim _{l\rightarrow \infty } u^l_j \le u_j\). Using this and arguments similar to those in the previous paragraph, we have
Letting \(k \rightarrow \infty \) gives
Recall that \(\{u_j > {\tilde{u}}_j\}\) is pluripolar. This coupled with the fact that T has no mass on pluripolar sets and Property (iii) of Proposition 3.5 yields
This finishes the proof. \(\square \)
Recall that for closed positive (1, 1)-currents \(R_1,R_2\) on X, we say that \(R_1\) is less singular than \(R_2\) if for every local chart U and psh function \(w_j\) on U such that \(R_j = \mathrm{dd}^cw_j\) on U for \(j=1,2\), then \(w_2 \le w_1+ O(1)\) on compact subsets of U; and \(R_1, R_2\) are of the same singularity type if \(w_1 \le w_2 + O(1)\) and \(w_2 \le w_1+ O(1)\) on compact subsets of U. The following generalizes [9, Proposition 2.1], see also [19, 22].
Proposition 4.2
Let X be a compact complex manifold of dimension n. Let m be an integer such that \(1 \le m \le n\). Let T be a closed positive current of bi-degree (p, p) on X. Let \(T_j, T'_j\) be closed positive (1, 1)-currents on X for \(1 \le j \le m\) such that \(T_j, T'_j\) are of the same singularity type and \(T_j= \mathrm{dd}^cu_j + \theta _j\), \(T'_j: =\mathrm{dd}^cu'_j+ \theta _j\), where \(\theta _j\) is a smooth form and \(u'_j, u_j\) are \(\theta _j\)-psh functions, for every \(1 \le j \le m\). Assume that for every \(J,J' \subset \{1, \ldots , m\}\) such that \(J \cap J' = \varnothing \), the product \(\langle \bigwedge _{j \in J}T_{j} \wedge \bigwedge _{j'\in J'} T'_{j'} {\dot{\wedge }} T \rangle \) is well defined. Then, for every \(\mathrm{dd}^c\)-closed smooth form \(\Phi \), we have
Proof
By compactness of X, we can assume \(u_j,u'_j \le 0\). By the hypothesis, \(\{u_j= -\infty \}= \{u'_j= -\infty \}\) and \(w_j:= u_j - u'_j\) is bounded outside \(\{u_j= -\infty \}\). Without loss of generality, we can assume that
outside \(\{u_j= -\infty \}\). Let \(A:= \cup _{j=1}^m \{u_j= -\infty \}\) which is a complete pluripolar set. Put \(u_{j k}:= \max \{u_j, -k\}\), \(u'_{j k}:= \max \{u'_j, -k\}\) and
which is quasi-psh and \(0\le \psi _k \le 1\), \(\psi _k(x)\) increases to 1 for \(x\not \in A\). We have \(\psi _k(x) =0\) if \(u_j(x)\le -k\) or \(u'_j(x) \le -k\) for some j. Put \(w_{j k}:= u_{j k} - u'_{j k}\). By (4.3), we have
on X. Let \(J, J' \subset \{1, \ldots , m\}\) with \(J \cap J' = \varnothing \) and
The last current is the difference of two closed positive \((|J|+|J'|+p, |J|+|J'|+p)\)-currents. Hence, \(R_{JJ'k}\) might not be positive in general and it is not clear how to control its mass as \(k \rightarrow \infty \). This is a subtle point which we need to pay attention to. The relative non-pluripolar product
exists by our assumption. Let
By Lemma 3.2, we get
for every \(J,J',k\). Put \({\tilde{R}}_{JJ'}:= \mathbf{1} _{X \backslash A} R_{JJ'}\). The last current is closed (for example see [4, Remark 1.10]) and positive because \(R_{JJ'} \ge 0\). Using the fact that \(\{\psi _{k} \not =0\} \subset B_{k} \backslash A\), we get
Claim. Let \(j'' \in \{1, \ldots , m\} \backslash (J \cup J')\). Let \(\Phi \) be \(\mathrm{dd}^c\)-closed smooth form of bi-degree \((p',p')\) on X, where \(p':=n- |J|-|J'|-p-1\). Then,
We prove Claim. Let \(\omega \) be a Hermitian metric on X. Let \(\eta : =\sum _{j=1}^m 2\, \theta _j\). We have \(\mathrm{dd}^c\psi _k+ k^{-1}\eta \ge 0\) for every k. By integration by parts and (4.6),
Observe that
because \(\mathrm{dd}^c\Phi = 0\) and \(\Phi \) is of bi-degree \((p',p')\). Write \(\mathrm{d}^c\Phi \) locally as a complex linear combination of forms like \(\tau _j \wedge \Phi _j\), where \(\tau _j\) is a (0, 1)-form or a (1, 0)-form and \(\Phi _j\) is a positive form. Hence, we can use the Cauchy–Schwarz inequality to obtain
where \(\Phi _0:= \omega ^{n- |J|- |J'|-1}\). We deduce that
by (4.5). Recall that \(\{\lim _{k \rightarrow \infty } \psi _k <1\}\) is equal to the complete pluripolar set A. Using this, Remark 2.7 and the fact that \({\tilde{R}}_{JJ'}\) has no mass on A, we get
(we recall that to get Remark 2.7 for \({\tilde{R}}_{JJ'}\) and \(\psi _k\), we need to use the strong quasi-continuity of \(\psi _k\) with respect to the capacity associated to \({\tilde{R}}_{JJ'}\)). Thus, we obtain
On the other hand, since
using similar arguments, we get
Combining this with (4.9) and (4.8) yields (4.7). Claim follows.
Now let \(S:= \langle T_{1} \wedge \cdots T_{n}{\dot{\wedge }} T \rangle - \langle T'_{1} \wedge \cdots T'_{n}{\dot{\wedge }} T \rangle \). Using \(T_{jk}= T'_{jk}+ \mathrm{dd}^cw_{jk}\), one can check that
This together with Claim yields \(\langle S, \Phi \rangle = \lim _{k\rightarrow \infty } \langle \psi _{k}S, \Phi \rangle =0\) for every \(\mathrm{dd}^c\)-closed smooth \(\Phi \). This finishes the proof. \(\square \)
Remark 4.3
Our proof of Proposition 4.2 still works if \(T'_j, T_j\) are not in the same cohomology class. In this case, one just needs to modify (4.2) accordingly.
The following result is the monotonicity property of relative non-pluripolar products mentioned in Introduction.
Theorem 4.4
Let X be a compact Kähler manifold of dimension n. Let \(T_j, T'_j\) be closed positive (1, 1)-currents on X for \(1 \le j \le m \) such that \(T_j, T'_j\) are in the same cohomology class for every j and \(T'_j\) is less singular than \(T_j\) for \(1 \le j \le m\). Let T be a closed positive current on X. Then, we have
Proof
Write \(T_j= \mathrm{dd}^cu_j + \theta _j\), \(T'_j= \mathrm{dd}^cu'_j + \theta _j\). Without loss of generality, we can assume that \(u'_j \ge u_j\). For \(l \in {\mathbb {N}}\), put \(u_{j}^l:= \max \{ u_j, u'_j - l \}\) which is of the same singularity type as \(u'_j\). Notice that \(\mathrm{dd}^cu_{j}^l+ \theta _j \ge 0\). Since X is Kähler, the current \(\langle \bigwedge _{j=1}^m (\mathrm{dd}^cu_{j}^l+ \theta _j) {\dot{\wedge }} T\rangle \) is of mass uniformly bounded in l. Let S be a limit current of \(\langle \bigwedge _{j=1}^m (\mathrm{dd}^cu_{j}^l+ \theta _j) {\dot{\wedge }} T\rangle \) as \(l \rightarrow \infty \). Since \(u_{j}^l\) decreases to \(u_j\) as \(l \rightarrow \infty \), we can apply Lemma 4.1 to get
Consequently, \(\{S\} \ge \{\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \}\). Using Proposition 4.2, we see that \(\{S\}\) is equal to \(\{\langle \bigwedge _{j=1}^m T'_j {\dot{\wedge }} T \rangle \}\). The desired assertion, hence, follows. This finishes the proof. \(\square \)
Remark 4.5
Let the notation be as in Theorem 4.4. Let T be of bi-degree (1, 1) and \(T'\) a closed positive (1, 1)-current which is less singular than T. Then, by using arguments similar to those in the proof of Theorem 4.4, we can prove that
(here we need to use Lemma 4.1 for a suitable sequence \((T_l)_l\) provided by Theorem 2.2). The inequality (4.11) offers us a way to define a notion of full mass intersection when T is of bi-degree (1, 1). This notion differs from those used below and in [4], albeit all of them are closely related. We will not go into details in this paper.
Consider, from now on, a compact Kähler manifold X with a Kähler form \(\omega \). Let T be a closed positive (p, p)-current on X. For every pseudoeffective (1, 1)-class \(\beta \) in X, we define its polar locus \(I_\beta \) to be that of a current with minimal singularities in \(\beta \). This is independent of the choice of a current with minimal singularities.
Let \(\alpha _1, \ldots ,\alpha _m\) be pseudoeffective (1, 1)-classes of X. Let \(T_{1, \min }, \ldots , T_{m,\min }\) be currents with minimal singularities in the classes \(\alpha _1, \ldots , \alpha _m\), respectively. By Theorem 4.4 and Lemma 3.4, the class \(\{\langle T_{1,\min } \wedge \cdots \wedge T_{m, \min } {\dot{\wedge }} T \rangle \}\) is a well-defined pseudoeffective class which is independent of the choice of \(T_{j, \min }\). We denote the last class by \(\{\langle \alpha _1\wedge \cdots \wedge \alpha _m {\dot{\wedge }} T \rangle \}\).
Proposition 4.6
-
(i)
The product \(\{\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}\) is symmetric and homogeneous in \(\alpha _1,\) \(\ldots , \alpha _m\).
-
(ii)
Let \(\alpha '_1\) are a pseudoeffective (1, 1)-class. Assume that T has no mass on \(I_{\alpha _1}\cup I_{\alpha '_1}\). Then, we have
$$\begin{aligned} \{\langle (\alpha _1+ \alpha '_1) \wedge \bigwedge _{j=2}^m \alpha _j {\dot{\wedge }} T \rangle \} \ge \{\langle \bigwedge _{j=1}^m \alpha _j\wedge T \rangle \}+ \{\langle \alpha '_1 \wedge \bigwedge _{j=2}^m \alpha _j {\dot{\wedge }} T \rangle \}. \end{aligned}$$ -
(iii)
Let \(1 \le l \le m\) be an integer. Let \(\alpha ''_1, \ldots , \alpha ''_l\) be a pseudoeffective (1, 1)-class such that \(\alpha ''_j \ge \alpha _j\) for \(1 \le j \le l\). Assume that T has no mass on \(I_{\alpha ''_j- \alpha _j}\) for every \(1 \le j \le l\). Then, we have
$$\begin{aligned} \{\langle \bigwedge _{j=1}^l \alpha ''_j \wedge \bigwedge _{j=l+1}^m \alpha _j {\dot{\wedge }} T \rangle \} \ge \{ \langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}. \end{aligned}$$ -
(iv)
If T has no mass on proper analytic subsets on X, then the product \(\{\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}\) is continuous on the set of \((\alpha _1,\ldots , \alpha _m)\) such that \(\alpha _1, \ldots , \alpha _m\) are big.
-
(v)
If T has no mass on proper analytic subsets on X and \(\alpha _1, \ldots , \alpha _m\) are big nef, then we have
$$\begin{aligned} \{\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}= \bigwedge _{j=1}^m \alpha _j \wedge \{T\}. \end{aligned}$$
We refer to [4, 5] for related statements in the case where T is the current of integration along X.
Proof
The desired assertion (i) follows from Proposition 3.5. We now prove (ii). Let \(T_{\min , \alpha _j}, T_{\min , \alpha '_j}\) be currents with minimal singularities in \(\alpha _j, \alpha '_j\), respectively. Observe that \(T_{\min ,\alpha _j}+ T_{\min , \alpha '_j}\) are in \((\alpha _j+ \alpha '_j)\). Thus, by Theorem 4.4, we get
The last class is equal to \(\{\langle \bigwedge _{j=1}^m T_{\min ,\alpha _j}{\dot{\wedge }} T \rangle \}+ \{\langle T_{\min , \alpha '_1} \wedge \bigwedge _{j=2}^m T_{\min ,\alpha _j} {\dot{\wedge }} T \rangle \}\) because of the hypothesis and Property (iv) of Proposition 3.5. Hence, (ii) follows. Similarly, we get (iii) by using Property (v) of Proposition 3.5.
We prove (iv). Observe that by a result of Demailly on analytic approximation of currents ([12]), the polar locus \(I_\beta \) of a big class \(\beta \) is contained in a proper analytic subset of X if \(\alpha \) is big. Using this, we see that (iv) is a direct consequence of (iii) and the observation that given a constant \(\epsilon >0\), for every \(\alpha '_j\) closed enough to \(\alpha _j\), we have that the classes \(\alpha _j' -(1-\epsilon ) \alpha _j\) and \( (1+ \epsilon ) \alpha _j- \alpha '_j\) are big (we use here the bigness of \(\alpha _j\)).
It remains to check (v). By (iv) and the bigness of \(\alpha _j\), we get
Since \(\alpha _j\) is nef, the limit in the left-hand side of the last equality is equal to \(\bigwedge _{j=1}^m \alpha _j \wedge \{T\}\). The proof is finished. \(\square \)
When T is the current of integration along X, we write \(\langle \alpha _1 \wedge \cdots \wedge \alpha _m \rangle \) for \(\{\langle \alpha _1\wedge \cdots \wedge \alpha _m {\dot{\wedge }} T \rangle \}\). We would like to comment that the class \(\langle \alpha _1\wedge \cdots \wedge \alpha _m \rangle \) is always bounded from above by the positive product of \(\alpha _1, \ldots , \alpha _m\) defined in [4, Definition 1.17]. They are equal if \(\alpha _1, \ldots , \alpha _m\) are big by Property (iv) of Proposition 4.6. However, we do not know if they are equal in general, even if \(\alpha _1, \ldots , \alpha _m\) are nef.
Question 1 Given nef classes \(\alpha _1, \ldots , \alpha _m\), is \(\langle \alpha _1 \wedge \cdots \wedge \alpha _m \rangle \) defined above equal to the positive product of \(\alpha _1, \ldots , \alpha _m\) introduced in [4]?
Let \(T_1, \ldots , T_m\) be closed positive (1, 1)-currents on X. By Theorem 4.4, we have \(\{\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \} \le \{\langle \bigwedge _{j=1}^m \{T_j \} {\dot{\wedge }} T\rangle \}\). The equality occurs if the masses of these two classes are equal. This is the reason for the following definition.
Definition 4.7
We say that \(T_1, \ldots , T_m\) are of full mass (non-pluripolar) intersection relative to T if we have \(\{\langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \} = \{\langle \bigwedge _{j=1}^m \{T_j \} {\dot{\wedge }} T\rangle \}\).
When T is the current of integration along X, we simply say “full mass intersection” instead of “full mass intersection relative to T.” In the last case, we underline that this notion is the one given in [4] if \(\{T_1\}, \ldots , \{T_m\}\) are big. In general, if \(T_1, \ldots , T_m\) are of full mass intersection in the sense of [4], then they are so in the sense of Definition 4.7. However, we do not know whether the reversed statement holds.
Let \({\mathcal {E}}(\alpha _1,\ldots , \alpha _m,T)\) be the set of \((T_1, \ldots , T_m)\) such that \(T_j \in \alpha _j\) for \(1 \le j \le m\) and
or equivalently
Several less general versions of full mass intersections were introduced in [4, 9, 17]. Note that if \(T_j= \mathrm{dd}^cu_j + \theta _j\) for some smooth Kähler form \(\theta _j\) and \(u_j\) a \(\theta _j\)-psh function for every j, then \(T_1,\ldots , T_m\) are of full mass intersection relative to T if and only if
as \(k \rightarrow \infty \), where \(T_{jk}:= \mathrm{dd}^c\max \{u_j, -k\}+ \theta _j\) for \(1 \le j \le m\).
The following result tells us that currents with full mass intersection satisfy the convergence along decreasing or increasing sequences of potentials.
Theorem 4.8
Let X be a compact Kähler manifold of dimension n. Let \(T_{jl}\) be a closed positive (1, 1)-current for \(1 \le j \le m,\) \(l \in {\mathbb {N}}\) such that \(T_{jl}\) converges to \(T_j\) as \(l \rightarrow \infty \). Let \((U_s)_s\) be a finite covering of X by open subsets such that \(T_{jl} = \mathrm{dd}^cu_{jl,s}\), \(T_j= \mathrm{dd}^cu_{j,s}\) on \(U_s\) for every s. Assume that
-
(i)
\(T_1, \ldots , T_m\) are of full mass intersection relative to T,
-
(ii)
\(\{\langle \bigwedge _{j=1}^m\alpha _{jl}{\dot{\wedge }} T \rangle \}\) converges to \(\{\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}\) as \(l \rightarrow \infty \), where \(\alpha _{jl}:= \{T_{jl}\}\), \(\alpha := \{T_j\}\) and one of the following two conditions hold:
-
(iii)
\(u_{jl,s} \ge u_{j,s}\) and \(u_{jl,s}\) converges to \(u_{j,s}\) in \(L^1_{loc}\) as \(l \rightarrow \infty \),
-
(iii’)
\(u_{jl,s}\) increases to \(u_{j,s}\) as \(l \rightarrow \infty \) almost everywhere and T has no mass on pluripolar sets.
Then we have
as \(l \rightarrow \infty \).
Proof
Observe that \(\langle \bigwedge _{j=1}^m T_{jl} {\dot{\wedge }} T \rangle \) is of uniformly bounded mass in l by (ii). Let S be a limit current of \(\langle \bigwedge _{j=1}^m T_{jl} {\dot{\wedge }} T \rangle \) as \(l \rightarrow \infty \). By (ii) again, we get \(\{S\} \le \{\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \}\). By Lemma 4.1 and Condition (iii) or \((iii')\), we have
By (i), the current in the right-hand side is of mass equal to \(\langle \bigwedge _{j=1}^m \alpha _j {\dot{\wedge }} T \rangle \) which is greater than or equal to the mass of S. So we get the equality \(S= \langle \bigwedge _{j=1}^m T_{j} {\dot{\wedge }} T \rangle \). The proof is finished. \(\square \)
Note that we do not require that \(T_{1l}, \ldots , T_{ml}\) are of full mass intersection relative to T. However, if \(T_{jl}, T_j\) are in the same cohomology class for every j, l, then by Theorem 4.4, Conditions (iii) and (i) of Theorem 4.8 imply that \(T_{1l}, \ldots , T_{ml}\) are of full mass intersection relative to T for every l. We notice here certain similarity of this result with [14, Proposition 5.4], where a notion of full mass intersection was considered for intersections of currents with analytic sets.
Here are some basic properties of currents with relative full mass intersection.
Lemma 4.9
Let X be a compact Kähler manifold. Let \(T_1, \ldots , T_m,T\) be closed positive currents on X such that \(T_j\) is of bi-degree (1, 1) and the cohomology class of \(T_j\) is Kähler for every j. Then, if \(T_1, \ldots , T_m\) are of full mass intersection relative to T, then the following three properties hold.
-
(i)
T has no mass on \(\cup _{j=1}^m I_{T_j}\).
-
(ii)
Let \(T'_j\) be closed positive (1, 1)-currents whose cohomology class are Kähler for \(1 \le j \le m\) such that \(T'_j\) is less singular than \(T_j\) for \(1 \le j \le m\). Then \(T'_1, \ldots , T'_m\) are of full mass intersection relative to T.
-
(iii)
For every subset \(J=\{j_1, \ldots , j_{m'}\} \subset \{1, \ldots , m\}\), the currents \(T_{j_1}, \ldots , T_{j_{m'}}\) are of full mass intersection relative to T. Moreover, if T is the current of integration along X, then \(T_j\) has no mass on \(I_{T_j}\) for \(1 \le j \le m\). Moreover, we have
-
(iv)
\(T_1, \ldots , T_m\) are of full mass intersection relative to T if and only if \(T_1+C \omega , \ldots , T_m+C \omega \) are of full mass intersection relative to T for every constant \(C>0\).
Proof
The desired property (i) is a direct consequence of (vii) of Proposition 3.5 and the fact that \(\{T_j\}\) is Kähler for every j. Note that if \(\{T'_j\}=\{T_j\}\) for every j, then (ii) is a direct consequence of the monotonicity of relative non-pluripolar products (Theorem 4.4). The first claim of the desired property (iii) is deduced from the last assertion applies to \(T'_j:= T_j\) for \(j \in J\) and \(T'_j:= \theta _j\) for \(j \not \in J\), where \(\theta _j\) is a smooth Kähler form in the class \(\{T_j\}\). In particular, we have \(\{\langle T_j \rangle \} = \{T_j\}\). Recall that \(\langle T_j \rangle =\mathbf{1} _{X \backslash I_{T_j}} T_j\) (Proposition 3.6). Hence,
It follows that \(\mathbf{1} _{I_{T_j}} T_j=0\) or equivalently, \(T_j\) has no mass on \(I_{T_j}\). Hence, (iii) follows.
Now observe that (iv) is a direct consequence of (iii), Property (iv) of Proposition 3.5 and (i). Finally, when \(\{T'_j\} \not = \{T_j\}\), Property (iv) allows us to add to \(T_j,T'_j\) suitable Kähler forms such that they are in the same cohomology class. So (ii) follows. The proof is finished. \(\square \)
In the last part of this section, we study Lelong numbers of currents of relative full mass intersection.
Lemma 4.10
Let V be an irreducible analytic subset of X. Let u be a quasi-psh function on X such that the generic Lelong number of u along V is strictly positive. Let v be a quasi-psh function on X having logarithmic poles along V, i.e., locally near V, we have
where \(\lambda >0\) is a constant, and \((f_j)_j\) is a local system of generators of the ideal sheaf associated to V. Then there exists a constant \(c >0\) such that \(u \le c v+ O(1)\) on X.
Note that given an analytic set V in X, there always exists a quasi-psh function v as in the hypothesis of Lemma 4.10; see [13, Lemma 2.1].
Proof
We can assume that u, v are \(\omega \)-psh functions. Put \(T:= \mathrm{dd}^cu+\omega \). Note that by hypothesis and Siu’s semi-continuity theorem, there is a constant \(c_0>0\) such that \(\nu (u,x)\ge c_0>0\) for every \(x \in V\) and \(\nu (u,x)= c_0\) for generic x in V. Consider first the case where V is of codimension 1. Thus, \(\mathbf{1} _{V}T= c_0 [V]\) is a nonzero positive current ([12, Lemma 2.17]) and \(c_1 v_1 + O(1) \le v \le c_1 v_1+ O(1)\) for some constant \(c_1>0\), where \(v_1\) is a potential of [V]. Since \(T \ge \mathbf{1} _{V}T= c_0[V]\), we get \(u \le c_0 v_1+O(1)\) on X. Thus, \(u \le (c_0/c_1) \, v + O(1)\) on X.
Consider now \(\mathrm{codim\ \!}V \ge 2\). By principalizing the ideal sheaf associated to V, we obtain a compact Kähler manifold \(X'\) and a surjective map \(\rho : X' \rightarrow X\) such that \(V':=\rho ^{-1}(V)\) is a hypersurface. Note that \(v\circ \rho \) is a quasi-psh function having logarithmic poles along an effective divisor supported on \(V'\). Applying the first part of the proof gives the desired assertion. This finishes the proof.
\(\square \)
Recall for every psh function v, the unbounded locus L(v) of v defined to be the set of x such that v is unbounded in every open neighborhood of x. Observe that if v has analytic singularities along V, then \(L(v)=V\). For every closed positive (1, 1)-current T, we define L(T) to be the unbounded locus of a potential of T. Here is a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers.
Theorem 4.11
Let X be a compact Kähler manifold. Let \(T_1, \ldots , T_m\) be closed positive (1, 1)-currents on X such that the cohomology class of \(T_j\) is Kähler for \(1 \le j \le m\) and T a closed positive (p, p)-current on X with \(p+m \le n\). Let V be an irreducible analytic subset in X such that the generic Lelong numbers of \(T_1, \ldots , T_m,T\) are strictly positive. Assume \(T_1, \ldots , T_m\) are of full mass intersection relative to T. Then, we have \(\dim V <n-p-m\).
Proof
Since \(\theta _j\) is Kähler for every j, we can use Theorem 4.4, Lemma 4.10 and the comment following it to reduce the setting to the case where \(T_j\) are currents with analytic singularities along V, for \(1 \le j \le m\) (hence \(L(T_j)= V\)). Since T is of bi-degree (p, p) and T has positive Lelong number everywhere on V, we deduce that the dimension of V is at most \(n-p\). Let s be a nonnegative integer such that \(\dim V = n-p -s\). We need to prove that \(s >m\). Suppose on the contrary that \(s \le m\). By Lemma 4.9, the currents \(T_1,\ldots , T_s\) are of full mass intersection relative to T.
Since \(L(T_j)= V\) for every j, we see that for every \(J \subset \{1, \ldots , s\}\), the Hausdorff dimension of \(\cap _{j \in J}L(T_j) \cap \mathrm{Supp}T\) is less than or equal to that of \(\cap _{j \in J}L(T_j)\) which is equal to \(\dim V= n-p- s \le n-p - |J|\). Thus, by [11, 16], the intersection of \(T_1, \ldots , T_s, T\) is classically well defined. By a comparison result on Lelong numbers ([11, Page 169]) and the fact that \(T_1, \ldots , T_m,T\) have strictly positive Lelong number at every point in V, we see that the Lelong number of \(\bigwedge _{j=1}^sT_j \wedge T\) at every point of V is strictly positive. Thus, the current \(\bigwedge _{j=1}^sT_j \wedge T\) has strictly positive mass on V (see [12, Lemma 2.17]). So by Proposition 3.6, \(\langle \bigwedge _{j=1}^sT_j \wedge T \rangle \) is not of maximal mass. This is a contradiction. This finishes the proof. \(\square \)
5 Weighted class of currents of relative full mass intersection
In this section, we introduce the notion of weighted classes of currents with full mass intersection relative to a closed positive current T. We only consider convex weights in the sequel. We refer to [4, 7, 17] for the case where T is the current of integration along X and information about non-convex weights. We also note that [4] considers currents in big classes whereas our setting here is restricted to currents in Kähler classes.
We fix our setting. Let \(0 \le p \le n\) be an integer and let T be a closed positive current of bi-degree (p, p) on X. For Kähler classes \(\alpha _1,\ldots , \alpha _m\) (\(m \le n-p\)) on X, we denote by \({\mathcal {E}}(\alpha _1,\ldots , \alpha _m, T)\) the set of m-tuple \((T_1,\ldots , T_m)\) of closed positive (1, 1)-currents on X such that \(T_j \in \alpha _j\) for \(1 \le j \le m\), and \(T_1,\ldots , T_m\) are of full mass intersection relative to T (see Definition 4.7).
Let \(\omega \) be a Kähler form on X. Let \(m \in {\mathbb {N}}^*\). Let \(T_j= \mathrm{dd}^cu_j + \theta _j\) be a closed positive (1, 1)-current where \(\theta _j\) is a smooth Kähler form and \(u_j\) is a negative \(\theta _j\)-psh function for \(1 \le j \le m\). We assume \(p+m \le n\). This is a minimal assumption because otherwise the relative non-pluripolar product is automatically zero by a bi-degree reason.
For \(k \in {\mathbb {N}}\), put \(u_{jk}:= \max \{u_j, -k\}\) and \(T_{jk}:= \mathrm{dd}^cu_{jk}+ \theta _j\). Note that \(T_{jk} \ge 0\) because \(\theta _j \ge 0\). Let \(\beta \) be a Kähler (1, 1)-class and \({\mathcal {E}}_m(\beta ,T)\) the set of closed positive (1, 1)-currents P in the class \(\beta \) such that \((P, \ldots , P)\) (m times P) is in \( {\mathcal {E}}(\beta , \ldots , \beta , T)\) (m times \(\beta \)). The class \({\mathcal {E}}_n(\beta )\) was introduced in [17]. We first prove the following result giving the convexity of the class of currents of relative full mass intersection.
Theorem 5.1
Assume that \(T_j, \ldots , T_j\) (m times \(T_j\)) are of full mass intersection relative to T for \(1 \le j \le m\). Then, \(T_1, \ldots , T_m\) are also of full mass intersection relative to T. In particular, for Kähler (1, 1)-classes \(\beta , \beta '\), we have
and \({\mathcal {E}}_{m}(\beta ,T)\) is convex .
Proof
Note that the second desired assertion is a direct consequence of the first one (recall that \(\{T_j\}\)’s are Kähler). By Lemma 4.9, T has no mass on \(I_{T_j}\) for every \(1 \le j \le m\). The first desired assertion follows from the following claim.
Claim. Let \(P_j\) be one of currents \(T_{1}, \ldots , T_{m}\) for \(1 \le j \le m\). Then, the currents \(P_1, \ldots , P_{m}\) are of full mass intersection relative to T.
Let \({\tilde{m}}\) be an integer such that there are at least \({\tilde{m}}\) currents among \(P_{1}, \ldots , P_{m}\) which are equal. We have \(0 \le {\tilde{m}} \le m\). We will prove Claim by induction on \({\tilde{m}}\). When \({\tilde{m}}=m\), the desired assertion is clear by the hypothesis. Assume that it holds for every \({\tilde{m}}'> {\tilde{m}}\). We need to prove it for \({\tilde{m}}\). By Lemma 4.9, we can assume that \(\theta _j\)’s are all equal to a form \(\theta \).
Without loss of generality, we can assume that \(P_{j}=T_1\) for every \(1 \le j \le {\tilde{m}}\) and \(P_{({\tilde{m}}+1)}= T_2\). If \(P_j= T_l\), then we define \(v_{jk}:= u_{lk}\) and \(P_{jk}:= T_{lk}\). Put
and \({\tilde{P}}:= \mathrm{dd}^c\max \{u_{1}, u_{2}\}+ \theta \). Since the cohomology class of \(T_j\) is Kähler for every j, we can apply (4.12) to \(P_j\). Hence, in order to obtain the desired assertion, we need to check that
as \(k,l \rightarrow \infty \), where
By Theorem 2.9, we get
This implies
It follows that
On the other hand, by induction hypothesis, the currents \(T_1,\ldots T_1, P_{{\tilde{m}}+2}, \ldots , P_{m}\) (\(({\tilde{m}}+1)\) times \(T_1\)) are of full mass intersection relative to T. This combined with Theorem 4.4 implies that \(T_1,\ldots T_1, {\tilde{P}}, P_{{\tilde{m}}+2}, \ldots , P_{m}\) (\({\tilde{m}}\) times \(T_1\)) are of full mass intersection relative to T because \({\tilde{P}}\) is less singular than \(T_1\). Using this, (4.12) and the fact that
we see that the right-hand side of (5.2) converges to 0 as \(k \rightarrow \infty \). It follows that
as \(k \rightarrow \infty \).
Now let \({\tilde{P}}'_k:= \mathrm{dd}^c\max \{u_{1k}, u_{2k}-1\}+ \theta \). The last current converges to
Observe that \({\tilde{P}}'\) is less singular than \(T_2\). By induction hypothesis and Theorem 4.4 as above, we see that the currents \({\tilde{P}}', \ldots , {\tilde{P}}', T_2, P_{{\tilde{m}}+2}, \ldots , P_{m}\) (\({\tilde{m}}\) times \({\tilde{P}}'\)) are of full mass intersection relative to T. Moreover, since T has no mass on \(I_{T_1}\), by (iii) of Proposition 3.5, we get
Using similar arguments as in the first part of the proof, we obtain
because \({\tilde{P}}', \ldots , {\tilde{P}}', T_2, P_{{\tilde{m}}+2}, \ldots , P_{m}\) (\({\tilde{m}}\) times \({\tilde{P}}'\)) are of full mass intersection relative to T. Observe that the right-hand side of (5.5) converges to
as \(k \rightarrow \infty \). The last quantity is equal to 0 because of (5.4). Consequently, we get
as \(k \rightarrow \infty \). Combining this and (5.3) and the fact that \(X= \{u_{1k} < u_{2k}\} \cup \{u_{1k} > u_{2k}-1\}\) gives (5.1). The proof is finished. \(\square \)
Let \(\chi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a continuous increasing function such that \(\chi (-\infty )= -\infty \) and for every constant c, there exist constants \(M,c_1, c_2>0\) so that
for \(t< -M\). Such a function is called a weight. For every function \(\xi \) bounded from above on X, let
The following simple lemma, which generalizes [17, Proposition 1.4], will be useful in practice.
Lemma 5.2
Let \(\xi := \chi (\sum _{j=1}^m u_j)\) and \(\xi _k:= \chi (\max \{\sum _{j=1}^m u_{j}, -k\})\). Assume that the current \(T_1, \ldots , T_m\) are of full mass intersection relative to T. Then we have
which increases to \({\tilde{E}}_\xi (T_1, \ldots ,T_m,T)\) as \(k \rightarrow \infty \); and if additionally \({\tilde{E}}_\xi (T_1, \ldots ,T_m,T)< \infty \), then
as \(k \rightarrow \infty \).
Proof
Put \(Q:= \langle \bigwedge _{j=1}^m T_j {\dot{\wedge }} T \rangle \) and \(Q_k:= \bigwedge _{j=1}^m T_{jk} \wedge T\). Let \(\xi ^0_k\) be the value of \(\xi _k\) on \(\cup _{j=1}^m \{u_j \le -k\}\). We have
Hence (5.8) follows by integrating the last equality over X and using the hypothesis. We check (5.9). Since \(Q_k=Q\) on \(\cap _{j=1}^m \{u_j >-k\}\), we have
On the other hand, observe that
as \(k \rightarrow \infty \) because Q has no mass on \(\cup _{j=1}^m \{u_j = -\infty \}\). Hence, \(\Vert \mathbf{1} _{\cup _{j=1}^m \{u_j \le -k\}} \xi _k Q \Vert \rightarrow 0\) as \(k \rightarrow \infty \). Observe
converging to 0 as \(k \rightarrow \infty \) by (5.8). Thus, (5.9) follows. The proof is finished. \(\square \)
Definition 5.3
We say that \(T_1, \ldots , T_m\) are of full mass intersection relative to T with weight \(\chi \) if \(T_1, \ldots , T_m\) are of full mass intersection relative to T and for every nonempty set \(J \subset \{1, \ldots , m\}\), we have \({\tilde{E}}_\xi \big ((T_j)_{j \in J},T\big )< \infty \), where \(\xi := \chi (\sum _{j=1}^m u_j)\) .
The last definition is independent of the choice of potentials \(u_j\) by (5.6). For currents \(T_1, \ldots , T_m\) of full mass intersection relative to T and \(\xi := \chi (\sum _{j=1}^m u_j)\), we can also define the joint \(\xi \)-energy relative to T of \(T_1, \ldots , T_m\) by putting
where the sum is taken over every subset J of \(\{1, \ldots , m\}\). The last energy depends on the choice of potentials but its finiteness does not.
Let \(\alpha _1, \ldots , \alpha _m\) be Kähler (1, 1)-classes on X. Let \({\mathcal {E}}_{\chi }(\alpha _1, \ldots , \alpha _m, T)\) be the set of m-tuple \((T_1, \ldots ,T_m)\) of closed positive (1, 1)-currents such that \(T_1, \ldots , T_m\) of full mass intersection relative to T with weight \(\chi \) and \(T_j \in \alpha _j\) for \(1 \le j \le m\).
For pseudoeffective (1, 1)-class \(\beta \), we define \({\mathcal {E}}_{\chi , m}(\beta , T)\) to be the subset of \({\mathcal {E}}_m (\beta , T)\) consisting of P such that \((P, \ldots , P)\) is in \({\mathcal {E}}_\chi (\beta , \ldots , \beta , T)\) (m times \(\beta \)). When \(m=n\), the class \({\mathcal {E}}_{\chi ,m}(\beta , T)\) was mentioned in [17, Section 5.2.2]. The last class was studied in [4, 17] when T is the current of integration along X and \(m=n\).
For \(P \in {\mathcal {E}}_{\chi , m}(\beta , T)\) with \(P= \mathrm{dd}^cu+ \theta \) (\(\theta \) is Kähler) and \(\xi := \chi (m\, u)\), we put
(m times P). The following result explains why our weighted class generalizes that given in [17].
Lemma 5.4
-
(i)
A current \(P \in {\mathcal {E}}_{\chi , m}(\beta , T)\) if and only if \(P \in {\mathcal {E}}_m(\beta ,T)\) and \(\xi \) is integrable with respect to \(\langle P^m \wedge T \rangle \).
-
(ii)
Assume that T is the current of integration along X and \(m=n\). Then, the energy \(E_{\chi , m}(P,T)\) is equivalent to the energy associated to \({\tilde{\chi }}(t):= \chi (n \, t)\) given in [17].
Proof
This is a direct consequence of computations in the proof of [4, Proposition 2.8 (i)] and Lemma 5.2: one first consider the case where \(\chi \) is smooth and use Lemma 5.7 below; the general case follows by regularizing \(\chi \). The proof is finished. \(\square \)
The following result is obvious.
Lemma 5.5
Let \(\eta _j\) be a positive closed (1, 1)-form for \(1 \le j \le m\). Then \(T_1, \ldots , T_m\) are of full mass intersection relative to T with weight \(\chi \) if and only if \((T_1+ \eta _1), \ldots , (T_m+\eta _m),T\) are of full mass intersection relative to T with weight \(\chi \).
From now on, we focus on convex weights. Let \({\mathcal {W}}^-\) be the set of convex increasing functions \(\chi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) with \(\chi (-\infty )= - \infty \). Observe that if \(\chi \in {\mathcal {W}}^- \), then \(\chi \) is automatically continuous. Basic examples of \(\chi \) are \(-(-t)^r\) for \(0 < r \le 1\). We have the following important observation which, in particular, implies that \({\mathcal {W}}^-\) is a set of weights in the sense given above.
Lemma 5.6
Let \(\chi \in {\mathcal {W}}^-\). Let g be a smooth radial cut-off function on \({\mathbb {R}}\), i.e., \(g(t)= g(-t)\) for \(t \in {\mathbb {R}}\), g is of compact support, \(0 \le g \le 1\) and \(\int _{\mathbb {R}}g(t) dt =1\). Put \(g_\epsilon (t):= \epsilon ^{-1}g(\epsilon t)\) for every constant \(\epsilon >0\) and \(\chi _\epsilon := \chi * g_\epsilon \) (the convolution of\(\chi \) with \(g_\epsilon \)). Then, \(\chi _\epsilon \in {\mathcal {W}}^-\), \(\chi _\epsilon \ge \chi \) and
for \(t\le 0\). Consequently, (5.6) holds for \(\chi \in {\mathcal {W}}^- \), or in other words, \(\chi \) is a weight.
Clearly, we always have that \(\chi _\epsilon \) converges uniformly to \(\chi \) because of continuity of \(\chi \).
Proof
By definition,
We deduce that since \(\chi \) is convex, \(\chi _\epsilon \ge \chi \) for every \(\epsilon \). The inequality (5.10) is a direct consequence of the convexity. It remains to prove the last desired assertion. We can assume \(\chi \) is smooth by the previous part of the proof. Since \(\chi \) is increasing, it is enough to prove the desired assertion for \(c\ge 0\). Fix a constant \(c\ge 0\). Consider \(t < -|c|+M\) for some big constant M. Write \(\chi (t+c)= \chi (t)+ \int _t^{t+c} \chi '(r) dr\). Using (5.10), we obtain that there is a constant \(c_1>0\) independent of c such that
Thus the desired assertion follows. The proof is finished. \(\square \)
We will need the following computation which seems to be used implicitly in the literature.
Lemma 5.7
Let \(\chi \in \mathscr {C}^3({\mathbb {R}})\) and \(w_1, w_2\) bounded psh functions on an open subset U of \({\mathbb {C}}^n\). Let Q be a closed positive current of bi-dimension (1, 1) on U. Then we have
and the operator \(w_1 \mathrm{dd}^c\chi (w_2) \wedge Q\) is continuous (in the usual weak topology of currents) under decreasing sequences of smooth psh functions converging to \(w_1,w_2\). Consequently, if f is a smooth function with compact support in U, then the equality
holds. Moreover, for f as above, we also have
Proof
Clearly, all of three desired equalities follows from the integration by parts if \(w_1, w_2\) are smooth. The arguments below essentially say that both sides of these equalities are continuous under sequences of smooth psh functions decreasing to \(w_1, w_2\). This is slightly non-standard due to the presence of Q even when \(\chi \) is convex.
First observe that (5.12) is a consequence of the second desired assertion because both sides of (5.12) are continuous under a sequence of smooth psh functions decreasing to \(w_2\). We prove (5.11). The desired equality (5.11) clearly holds if \(w_2\) is smooth. In general, let \((w_2^\epsilon )_\epsilon \) be a sequence of standard regularizations of \(w_2\). Recall that \( \mathrm{dd}^c\chi (w_2) \wedge Q \) is defined to be \(\mathrm{dd}^c\big ( \chi (w_2)Q\big )\) which is equal to the limit of \(\mathrm{dd}^c\big ( \chi (w^\epsilon _2)Q\big )\) as \(\epsilon \rightarrow 0\). By (5.11) for \(w_2^\epsilon \) in place of \(w_2\), we see that \(\mathrm{dd}^c\big ( \chi (w^\epsilon _2)Q\big )\) is of uniformly bounded mass. As a result, \(\mathrm{dd}^c\chi (w_2) \wedge Q\) is of order 0. Thus, \(w_1 \mathrm{dd}^c\chi (w_2)\wedge Q\) is well defined. Put
Recall that \(I(1, w_2^\epsilon , w_2^\epsilon ) \rightarrow \mathrm{dd}^c\chi (w_2) \wedge Q\). By Corollary 2.5, we have
as \(\epsilon \rightarrow 0\). On the other hand, since \(\chi ''\) is in \(\mathscr {C}^1\), we get
This combined with the convergence of Monge–Ampère operators under decreasing sequences tells us that
as \(\epsilon \rightarrow 0\). Combining (5.15) and (5.14) gives that \(I(w_1,w_2^\epsilon , w_2^\epsilon ) \rightarrow I(w_1,w_2,w_2)\) as \(\epsilon \rightarrow 0\). Letting \(w_1\equiv 1\) in the last limit, we get (5.11). The second desired assertion also follows. We prove (5.13) similarly. The proof is finished. \(\square \)
Here is a monotonicity property of weighted classes.
Theorem 5.8
(Monotonicity of weighted classes) Let \(\chi \in {\mathcal {W}}^-\) with \(|\chi (0)| \le 1\). Let \(T'_j \) be a closed positive (1, 1)-current whose cohomology class is Kähler for \(1 \le j \le m\). Assume that \(T'_j\) is less singular than \(T_j\) and \(T_1, \ldots , T_m\) are of full mass intersection relative to T with weight \(\chi \). Then, \(T'_1, \ldots , T'_m\) are also of full mass intersection relative to T with weight \(\chi \) and for \(\xi := \chi \big (\sum _{j=1}^m u_j\big )\), we have
for some constants \(c_1,c_2>0\) independent of \(\chi \).
Proof
Let \(T'_j= \mathrm{dd}^cu'_j+ \theta '_j\) for some smooth Kähler forms \(\theta '_j\). Put \(v:= \sum _{j=1}^m u_j\), \(v_k:= \max \{v, -k\}\), \(\xi := \chi (v)\) and \(\xi _k:= \chi (v_k)\). Let \(u_{jk}:= \max \{u_j, -k\}\) , \(T_{jk}:= \mathrm{dd}^cu_{jk}+ \theta _j\). Define \(u'_{jk}, T'_{jk}\) similarly. We have \(u_j \le u'_j\). Observe that
by Lemma 4.9. Hence, in order to obtain the first desired assertion, it suffices to check (5.16) because \(- \chi (\sum _{j=1}^m u'_j) \le - \xi \). The desired inequality (5.16) follows by letting \(k \rightarrow \infty \) in the following claim and using Lemma 5.2 and (5.17).
Claim. For every \(J, J' \subset \{1, \ldots , m\}\) with \(J \cap J' = \varnothing \), we have
for some constants \(c_1, c_2\) independent of \(\chi \).
It remains to prove Claim now. We observe that it is enough to prove Claim for \(\chi \) smooth by Lemma 5.6. Consider, from now on, smooth \(\chi \). We can also assume that \(u_j,u'_j<-M\) for some big constant M.
We prove Claim by induction on \(|J'|\). When \(J'= \varnothing \), this is clear. Assume Claim holds for every \(J'\) with \(|J'|< m'\). We need to prove that it holds for \(J'\) with \(|J'|=m'\). Without loss of generality, we can assume that \( 1 \in J'\). Put
By integration by parts, we obtain
Denote by \(I_{k,1}, I_{k,2}\) the first and second terms in the right-hand side of the last equality. By induction hypothesis, we get
for some constants \(c_1, c_2>0\) depending only on the masses of \(T_j,T'_j\) for \(1 \le j \le m\).
It remains to treat \(I_{k,1}\). Put \(\theta := \sum _{j=1}^m \theta _j\). Note that \(\mathrm{dd}^cv_k + \theta \ge 0\). By Lemma 5.7, the current \((\mathrm{dd}^c\xi _k+ \chi '(v_k) \theta ) \wedge Q_k\) is positive. This combined with the inequality \(u_{1k} \le u'_{1k}\) gives
because
(see (5.10)). This combined with induction hypothesis gives
Hence, Claim follows. The proof is finished. \(\square \)
By the above proof, one can check that if we fix \(\theta _1, \ldots , \theta _m\) and \(\theta '_1, \ldots , \theta '_m\), then the constants \(c_1\) and \(c_2\) in (5.16) can be chosen to be independent of \(T_1, \ldots , T_m\). Here is a convexity property for weighted classes.
Theorem 5.9
Let \(\chi \in {\mathcal {W}}^-\). Assume that \(T_j, \ldots , T_j\) (m times \(T_j\)) are of full mass intersection relative to T with weight \(\chi \) for \(1 \le j \le m\). Then, \(T_1, \ldots , T_m\) are also of full mass intersection relative to T with weight \(\chi \). In particular, for Kähler (1, 1)-classes \(\beta , \beta '\), we have
and \({\mathcal {E}}_{\chi , m}(\beta ,T)\) is convex.
Proof
The second and third desired assertions are direct consequences of the first one. We prove the first desired assertion. Firstly, by Theorem 5.1, we have
By Lemma 5.5, we can assume that \(\theta _j= \theta \) for every \(1 \le j \le m\). The desired assertion is a consequence of the following claim.
Claim. Let \(P_j\) be one of currents \(T_{1}, \ldots , T_{m}\) for \(1 \le j \le m\). Then, the currents \(P_1, \ldots , P_{m}\) are of full mass intersection relative to T with weight \(\chi \).
Let \({\tilde{m}}\) be an integer such that there are at least \({\tilde{m}}\) currents among \(P_{1}, \ldots , P_{m}\) which are equal. We have \(0 \le {\tilde{m}} \le m\). We will prove Claim by induction on \({\tilde{m}}\). When \({\tilde{m}}=m\), the desired assertion is clear by the hypothesis. Assume that it holds for every number \({\tilde{m}}'> {\tilde{m}}\). We need to prove it for \({\tilde{m}}\).
Without loss of generality, we can assume that \(P_{j}=T_1\) for every \(1 \le j \le {\tilde{m}}\) and \(P_{({\tilde{m}}+1)}= T_2\). If \(P_j= T_l\), then we define \(v_{jk}:= u_{lk}\) and \(P_{jk}:= T_{lk}\). Put \(Q:= \bigwedge _{j={\tilde{m}}+2}^{m} P_{jk} \wedge T\wedge \omega ^{n-m-p}\), \({\tilde{P}}_{k}:= \mathrm{dd}^c\max \{u_{1k}, u_{2k}\}+ \theta \) and
By Lemma 5.2 and (5.19), we need to check that
for some constant C independent of k. Actually, we need to verify a stronger statement that for every \(J \subset \{1,\ldots , {\tilde{m}}\}\), then
is uniformly bounded in k, but the proof will be similar to that of (5.20).
Observe that \(X= \{u_{1k} < u_{2k}\} \cup \{u_{1k} > u_{2k}-1\}\). Using Theorem 2.9, we get
This combined with the fact that \( - \chi ({\tilde{m}} u_{1k} + u_{2k}+ w_k) \le - \chi \big (({\tilde{m}}+1) u_{1k}+ w_k\big )\) on \(\{u_{1k} < u_{2k}\}\) implies
which is \(\le \)
By the fact that \({\tilde{P}}_k\) is less singular than \(T_{1k}\) and Claim in the proof of Theorem 5.8, we see that the right-hand side of the last inequality is bounded uniformly in k because \(T_1,\ldots T_1, P_{{\tilde{m}}+2}, \ldots , P_{m}\) (\(({\tilde{m}}+1)\) times \(T_1\)) are of full mass intersection relative to T with weight \(\chi \) (this is a consequence of the induction hypothesis). It follows that
for some constant C independent of k. Similarly, for \({\tilde{P}}'_k:= \mathrm{dd}^c\max \{u_{1k}, u_{2k}-1\}+ \theta \), we have
for some constants \(c_1, c_2>0\) independent of k (we used (5.6) here). Using the induction hypothesis again and the fact that \({\tilde{P}}'_k\) is less singular that \(T_{2k}\), we obtain
for some constant C independent of k. Combining (5.22) and (5.21) gives (5.20). The proof is finished. \(\square \)
We now give a continuity property of weighted classes which generalizes the second part of [4, Theorem 2.17]) in the case where the cohomology classes of currents are Kähler.
Proposition 5.10
Let \(\chi \in {\mathcal {W}}^-\). Let \(T_{j,l}= \mathrm{dd}^cu_{j,l}+ \theta _j\) be closed positive (1, 1)-currents, for every \(1 \le j \le m\), \(l \in {\mathbb {N}}\) such that \(u_{j,l} \rightarrow u_j\) as \(l \rightarrow \infty \) in \(L^1\) and \(u_{j,l}\ge u_j\) for every j, l. Assume that \(T_1, \ldots , T_m\) are of full mass intersection relative to T with weight \(\chi \). Then, we have
as \(l \rightarrow \infty \).
Proof
By Theorem 5.8, the currents \(T_{1,l}, \ldots , T_{m,l}\) are of full mass intersection relative to T with weight \(\chi \) for every l. The desired convergence now follows by using arguments similar to those in the proof of [4, Theorem 2.17] (notice also Lemma 5.2 and the comment after Theorem 5.8). The proof is finished. \(\square \)
Remark 5.11
As in [17], we can check that \({\mathcal {E}}(\alpha _1, \ldots , \alpha _m, T)= \cup _{\chi \in {\mathcal {W}}^-}{\mathcal {E}}_\chi (\alpha _1, \ldots , \alpha _m, T)\) for Kähler classes \(\alpha _1, \ldots \alpha _m\) on X.
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Acknowledgements
The author would like to thank Hoang Chinh Lu for kindly answering him numerous questions about non-pluripolar products. He also thanks Tamás Darvas, Lucas Kaufmann and Tuyen Trung Truong for their comments. This research is supported by a postdoctoral fellowship of the Alexander von Humboldt Foundation.
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Vu, DV. Relative non-pluripolar product of currents. Ann Glob Anal Geom 60, 269–311 (2021). https://doi.org/10.1007/s10455-021-09780-7
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DOI: https://doi.org/10.1007/s10455-021-09780-7