Curvature and $L^p$ Bergman Spaces on Complex Submanifolds in $\pmb {{\mathbb {C}}^N}$

Abstract

Let M be a closed complex submanifold in ${\mathbb {C}}^N$ with the complete Kähler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle L over M and the associated $L^2$ cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems. As applications we obtain some rigidity results concerning growth of curvatures.

Appendix: A Monotonic Property

The following result is motivated by Gromov’s work [12].

Proposition 7

Let (L, h) be a Hermitian line bundle over M. Suppose there exists $\alpha >0$ such that $\mathrm{Tr}_g(\Theta _h) \le \alpha .$ Then for $0<p_1<p_2\le \infty$ we have

$$\begin{aligned} H^0_{L^{p_1}} (M,L)\subset H^0_{L^{p_2}} (M,L). \end{aligned}$$

Proof

We first show that for $0<p<\infty$,

$$\begin{aligned}n H^0_{L^{p}} (M,L) \subset H^0_{L^{\infty }} (M,L). \end{aligned}$$

For every $s\in H^0_{L^{p}} (M,L)$ we have

$$\begin{aligned} \frac{p}{2} \Delta \log |s|^2_h \ge -2p \mathrm{Tr}_g(\Theta _h)\ge -2p \alpha . \end{aligned}$$

Set $\tilde{g}= g+ idtd\bar{t}$, $t\in {{{\mathbb {C}}}}$. It turns out that $\psi :=\log |s|^p_h+ p\alpha |t|^2$ is subharmonic with respect to $\tilde{g}$ on $\widetilde{M}:=M\times {{{\mathbb {C}}}}$, so is $e^{\psi }$. Since $\widetilde{M}$ is a minimal submanifold in ${{{\mathbb {C}}}}^{N+1}={{\mathbb {R}}}^{2N+2}$, we infer from the mean-value inequality (cf. [5, Corollary 1.17]) that for every $x\in M$,

$$\begin{aligned} |s|^p_h(x) = e^{\psi (x,0)}\le & {} \frac{\int _{\widetilde{M}((x,0),1)} e^{\psi (y,t)} \mathrm{d}V_y \mathrm{d}V_t }{\mathrm{vol}_\mathrm{eucl}(B_1\subset {{\mathbb {R}}}^{2n+2})}\\\le & {} \mathrm{const}_n\cdot {\int _{M(x,1)\times {{\mathbb {D}}}} e^{\psi (y,t)} \mathrm{d}V_y \mathrm{d}V_t }\\\le & {} \mathrm{const}_n\cdot \int _{M(x,1)} |s|^p_h \mathrm{d}V \cdot \int _{{\mathbb {D}}} e^{p\alpha |t|^2} \mathrm{d}V_t\\\le & {} \mathrm{const}_{n} \cdot \frac{e^{p\alpha }}{p\alpha } \cdot \int _{M} |s|^p_h \mathrm{d}V, \end{aligned}$$

i.e., $\Vert s\Vert _{L^\infty }\le \mathrm{const}_{n,p,\alpha } \cdot \Vert s\Vert _{L^p}$.

Now suppose $s\in H^0_{L^{p_1}} (M,L)$. We have

$$\begin{aligned} \int _{M} |s|^{p_2}_h \mathrm{d}V \le \int _{M} |s|^{p_1}_h \mathrm{d}V \cdot \Vert s\Vert _{L^\infty }^{p_2-p_1} <\infty , \end{aligned}$$

i.e., $s\in H^0_{L^{p_2}} (M,L)$. $\square$

Corollary 6

Suppose there exists $\alpha >0$ such that

$$\begin{aligned} \mathrm{Scal}_g \ge -\alpha . \end{aligned}$$

Then $P_{m,L^p}(M)$ is non-decreasing in p.

Remark 10

Fix $p<2$ and $k\in {\mathbb {Z}}^+$ with $kp\ge 2$. Suppose we have

$$\begin{aligned} s_j\in H^0_{L^2}(M,K_M^{\otimes m_j})\subset H^0_{L^{kp}}(M, K_M^{\otimes m_j}),\ \ \ j=1,\ldots ,k. \end{aligned}$$

Then

$$\begin{aligned} s:=s_1\otimes \cdots \otimes s_k\in H^0(M,K_M^{\otimes m}), \ \ \ m=m_1+\cdots +m_k. \end{aligned}$$

Hölder’s inequality gives

$$\begin{aligned} \int _M |s|_{h^{\otimes m}}^{p} \mathrm{d}V\le \prod _{j=1}^k \Vert s_j\Vert _{L^{kp}}^{p}<\infty , \end{aligned}$$

i.e., $s\in H^0_{L^{p}}(M,K_M^{\otimes m})$. Combined with Proposition 7 we may produce many $L^p$ pluri-canonical sections from $L^2$ pluri-canonical sections.