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Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space

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Abstract

We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia [5] which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation.

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References

  1. Aldrovandi, E., Falqui, G.: Geometry of Higgs and Toda fields on Riemann surfaces. J. Geom. Phys. 17(1), 25–48 (1995)

    Article  MathSciNet  Google Scholar 

  2. Álvarez-Cónsul, L., García-Prada, O.: Hitchin-Kobayashi correspondence, quivers, and vortices. Comm. Math. Phys. 238(1–2), 1–33 (2003)

    Article  MathSciNet  Google Scholar 

  3. Banfield, D.: The geometry of coupled equations in gauge theory, D. Phil. Thesis, University of Oxford (1996)

  4. Baptista, J.M.: Moduli spaces of Abelian vortices on Kähler manifolds, arXiv:1211.0012

  5. Baraglia, D.: Cyclic Higgs bundles and the affine Toda equations. Geom. Dedicata 174, 25–42 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bradlow, S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Comm. Math. Phys. 135(1), 1–17 (1990)

    Article  MathSciNet  Google Scholar 

  7. Bradlow, S.B.: Special metrics and stability for holomorphic bundles with global sections. J. Diff. Geom. 33(1), 169–213 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Bryan, J.A., Wentworth, R.: The multi-monopole equations for Kähler surfaces. Turkish J. Math. 20(1), 119–128 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Dai, S., Li, Q.: On cyclic Higgs bundles. Math. Ann. 376(3–4), 1225–1260 (2020)

    Article  MathSciNet  Google Scholar 

  10. Doan, A.: Adiabatic limits and Kazdan-Warner equations. Calc. Var. Partial Differ. Equ. 57(5), 25 (2018). (Art. 124)

    Article  MathSciNet  Google Scholar 

  11. Dolgachev, I.: Lectures on invariant theory London, Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  12. Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ (1993). xii+157 pp. ISBN: 0-691-00049-2

  13. García-Prada, O., Oliveira, A.: Connectedness of Higgs bundle moduli for complex reductive Lie groups. Asian J. Math. 21(5), 791–810 (2017)

    Article  MathSciNet  Google Scholar 

  14. Guest, M.A., Ho, N.-K.: Kostant, Steinberg, and the Stokes matrices of the \(tt^\ast \)-Toda equations. Selecta Math. (N.S.) 25(3), 43 (2019). (Paper No. 50)

    Article  MathSciNet  Google Scholar 

  15. Guest, M.A., Lin, C.-S.: Nonlinear PDE aspects of the \(tt^\ast \) equations of Cecotti and Vafa. J. Reine Angew. Math. 689, 1–32 (2014)

    Article  MathSciNet  Google Scholar 

  16. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55(1), 59–126 (1987)

    Article  MathSciNet  Google Scholar 

  17. Hitchin, N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992)

    Article  MathSciNet  Google Scholar 

  18. Jaffe, A., Taubes, C.H.: Vortices and monopoles, Progress in Physics, 2. Birkhäuser, Boston, Mass. (1980). v+287 pp. ISBN: 3-7643-3025-2

  19. Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. of Math. 99, 14–47 (1974)

    Article  MathSciNet  Google Scholar 

  20. Kempf, G., Ness, L.: The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), pp. 233–243, Lecture Notes in Math., 732, Springer, Berlin (1979)

  21. King, A.D.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. 45(180), 515–530 (1994)

    Article  MathSciNet  Google Scholar 

  22. Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984. i+211 pp. ISBN: 0-691-08370-3.

  23. Konno, H.: The geometry of toric hyperkähler varieties, Toric topology, 241–260, Contemp. Math., 460, Amer. Math. Soc., Providence, RI (2008)

  24. Kostant, Bertram: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81, 973–1032 (1959)

    Article  MathSciNet  Google Scholar 

  25. Lübke, M., Teleman, A.: The Kobayashi-Hitchin correspondence, p. x+254. World Scientific Publishing Co., Inc, River Edge, NJ (1995). ISBN: 981-02-2168-1

  26. Mumford, D.: The red book of varieties and schemes, Second, expanded edition. Includes the Michigan lectures (1974) on curves and their Jacobians. With contributions by Enrico Arbarello. Lecture Notes in Mathematics, 1358, pp. x+306. Springer-Verlag, Berlin (1999). ISBN: 3-540-63293-X

  27. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, pp. xiv+292. Springer-Verlag, Berlin (1994). ISBN: 3-540-56963-4

  28. Mundet i Riera, I.: A Hitchin-Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math. 528, 41–80 (2000)

    MathSciNet  MATH  Google Scholar 

  29. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, pp. xii+132. American Mathematical Society, Providence, RI (1999). ISBN: 0-8218-1956-9

  30. Newstead, P.E.: Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 51, pp. vi+183. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi (1978). ISBN: 0-387-08851-2

  31. Simpson, C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1(4), 867–918 (1988)

    Article  MathSciNet  Google Scholar 

  32. Simpson, C.T.: Katz’s middle convolution algorithm. Pure Appl. Math. Q. 5(2, Special Issue: In honor of Friedrich Hirzebruch. Part 1), 781–852 (2009)

  33. Taubes, C.H.: Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations. Comm. Math. Phys. 72(3), 277–292 (1980)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to his supervisor Professor Ryushi Goto for fruitful discussions and encouragements. He also wishes to express his gratitude to anonymous referee for careful reading and helpful suggestions.

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Geometric invariant theory and the moment maps for linear torus actions

Geometric invariant theory and the moment maps for linear torus actions

We give a brief review of the relationship between the geometric invariant theory and the moment maps for linear torus actions. In particular, we clarify the relationship between condition (1.5) and the stability condition of the geometric invariant theory. General references for this section are [11, 20,21,22, 27, 29] and [30].

1.1 Notation

We first fix our notation. Let K be a closed connected subtorus of a real torus \(T^d:=\mathrm{U}(1)^d\) with the Lie algebra \(k\subseteq t^d\). We denote by \(\iota ^*: (t^d)^*\rightarrow k^*\) the dual map of the inclusion map \(\iota : k\rightarrow t^d\). Let \(u_1,\dots , u_d\) be a basis of \(t^d\) defined by

$$\begin{aligned} u_1:=&\,({\sqrt{-1}}, 0, \dots , 0), \\ u_2:=&\,(0, {\sqrt{-1}}, 0,\dots , 0), \\&\cdots \\ u_d:=&\,(0,\dots , 0,{\sqrt{-1}}). \end{aligned}$$

We denote by \(u^1, \dots , u^d\in (t^d)^*\) the dual basis of \(u_1, \dots , u_d\). Let \((\cdot , \cdot )\) be the metric on \(t^d\) and \((t^d)^*\) satisfying

$$\begin{aligned} (u_i, u_j)=(u^i, u^j)=\delta _{ij} \ \ \text {for all} \, i, j, \end{aligned}$$

where \(\delta _{ij}\) denotes the Kronecker delta. The diagonal action of \(T^d\) on \({{\mathbb {C}}}^d\) induces an action of K which preserves the Kähler structure of \({{\mathbb {C}}}^d\). Let \(\mu _K:{{\mathbb {C}}}^d\rightarrow k^*\) be a moment map for the action of K which is defined by

$$\begin{aligned} \langle \mu _K(z), v \rangle =\frac{1}{2}g_{{{\mathbb {R}}}^{2d}}({\sqrt{-1}}vz, z) \ \ \text {for} \, v\in k, \end{aligned}$$

where we denote by \(g_{{{\mathbb {R}}}^{2d}}(\cdot , \cdot )\) the standard metric of \({{\mathbb {C}}}^d\simeq {{\mathbb {R}}}^{2d}\), and by \(\langle \cdot , \cdot \rangle \) the natural coupling. The moment map \(\mu _K\) is also denoted as

$$\begin{aligned} \mu _K(z)=-\frac{1}{2}\sum _{j=1}^d\iota ^*u^j |z_j|^2 \ \ \text {for} \, z=(z_1,\dots , z_d)\in {{\mathbb {C}}}^d. \end{aligned}$$

Let \(T^d_{{\mathbb {C}}}:=({{\mathbb {C}}}^*)^d\) be the complexification of \(T^d\). We define the exponential map \(\mathrm{Exp}: t^d\oplus {\sqrt{-1}}t^d\rightarrow T^d_{{\mathbb {C}}}\) by

$$\begin{aligned} \mathrm{Exp}(v+{\sqrt{-1}}v^\prime )=(e^{{\sqrt{-1}}\langle v+{\sqrt{-1}}v^\prime , u^1\rangle }, \dots , e^{{\sqrt{-1}}\langle v+{\sqrt{-1}}v^\prime , u^d\rangle }). \end{aligned}$$

We denote by \(K_{{\mathbb {C}}}\) the complexification of K. Let \(k_{{\mathbb {Z}}}\subseteq k\) be \(\ker \mathrm{Exp}|_k\) and \((k_{{\mathbb {Z}}})^*\) the dual. Note that \((k_{{\mathbb {Z}}})^*\) is naturally identified with \(\sum _{j=1}^d{{\mathbb {Z}}}\ (\iota ^*u^j/2\pi )\). For each \(\alpha \in (k_{{\mathbb {Z}}})^*\), we define a character \(\chi _\alpha : K_{{\mathbb {C}}}\rightarrow {{\mathbb {C}}}^*\) by

$$\begin{aligned} \chi _\alpha (\mathrm{Exp}(v+{\sqrt{-1}}v^\prime ))=e^{2\pi {\sqrt{-1}}\langle v+{\sqrt{-1}}v^\prime , \alpha \rangle }. \end{aligned}$$

1.2 Symplectic and GIT quotients

Let \(\alpha \in (k_{{\mathbb {Z}}})^*\). We define an action of \(K_{{\mathbb {C}}}\) on \({{\mathbb {C}}}^d\times {{\mathbb {C}}}\) by

$$\begin{aligned} g\cdot (z, v):=(gz, \chi _\alpha (g)^{-1}v) \ \ \text {for} \, (z, v)\in {{\mathbb {C}}}^d\times {{\mathbb {C}}}. \end{aligned}$$

Let \(R_\alpha \) be the invariant ring for the above action:

$$\begin{aligned} R_\alpha :=\{{\hat{f}}(x, y)\in {{\mathbb {C}}}[x_1,\dots , x_d, y]\mid {\hat{f}}(g\cdot (x,y))={\hat{f}}(x,y) \, \text {for all} \, g\in K_{{\mathbb {C}}}\}. \end{aligned}$$

By a theorem of Nagata, \(R_\alpha \) is finitely generated. For each \(n\in {{\mathbb {Z}}}_{\ge 0}\) let \(R_{\alpha , n}\) be a space of polynomials defined by

$$\begin{aligned} R_{\alpha , n}:=\{f(x)\in {{\mathbb {C}}}[x_1,\dots , x_d]\mid f(gx)=\chi _\alpha (g)^nf(x) \, \text {for all} \, g\in K_{{\mathbb {C}}}\}. \end{aligned}$$

Then \(R_\alpha \) is naturally identified with \(\bigoplus _{n\ge 0}R_{\alpha , n}\).

Definition 3

Define \({{\mathbb {C}}}^d//_\alpha K_{{\mathbb {C}}}:=\mathrm{Proj}(\bigoplus _{n\ge 0}R_{\alpha , n})\). This is called the geometric invariant theory (GIT) quotient.

Definition 4

We say \(z\in {{\mathbb {C}}}^d\) is \(\alpha \)-semistable if there exists an \(f(x)\in R_{\alpha , n}\) with \(n\in {{\mathbb {Z}}}_{>0}\) such that \(f(z)\ne 0\). We denote by \(({{\mathbb {C}}}^d)^{\alpha -ss}\) the set of all \(\alpha \)-semistable points.

We refer the reader to [26] for a proof of the following Proposition 1:

Proposition 1

Let V be a complex vector space and \(G\subseteq \mathrm{GL}(V)\) an algebraic subgroup. Then we have the following:

$$\begin{aligned} \overline{\overline{G\cdot p}}=\overline{G\cdot p} \ \ \text {for all} \, p\in V, \end{aligned}$$

where we denote by \(\overline{\overline{G\cdot p}}\) the Euclidean closure, and by \(\overline{G\cdot p}\) the Zariski closure. In particular, \(G\cdot p\) is closed with respect to the Euclidean topology if and only if it is closed with respect to the Zariski topology.

The GIT quotient can be described as follows:

Proposition 2

There exists a categorical quotient \(\phi :({{\mathbb {C}}}^d)^{\alpha -ss}\rightarrow {{\mathbb {C}}}^d//_\alpha K_{{\mathbb {C}}}\) which satisfies the following properties. For each \(z, z^\prime \in ({{\mathbb {C}}}^d)^{\alpha -ss}\), \(\phi (z)=\phi (z^\prime )\) holds if and only if \(\overline{K_{{\mathbb {C}}}\cdot z}\cap \overline{K_{{\mathbb {C}}}\cdot z^\prime }\cap ({{\mathbb {C}}}^d)^{\alpha -ss}\ne \emptyset \) and further for each \(q\in {{\mathbb {C}}}^d //_\alpha K_{{\mathbb {C}}}\), \(\phi ^{-1}(q)\) contains a unique \(K_{{\mathbb {C}}}\)-orbit which is closed in \(({{\mathbb {C}}}^d)^{\alpha -ss}\).

Proof

See [11, 27] and [30]. \(\square \)

We define an equivalence relation \(\sim \) on \(({{\mathbb {C}}}^d)^{\alpha -ss}\) as follows:

$$\begin{aligned} z\sim z^\prime \Longleftrightarrow \overline{K_{{\mathbb {C}}}\cdot z}\cap \overline{K_{{\mathbb {C}}}\cdot z^\prime }\cap ({{\mathbb {C}}}^d)^{\alpha -ss}\ne \emptyset \ \ \text {for} \, z, z^\prime \in ({{\mathbb {C}}}^d)^{\alpha -ss}. \end{aligned}$$

Then by Proposition 2, \({{\mathbb {C}}}^d//_\alpha K_{{\mathbb {C}}}\) is identified with \(({{\mathbb {C}}}^d)^{\alpha -ss}/\sim \). Moreover for each equivalent class there exists a \(z\in ({{\mathbb {C}}}^d)^{\alpha -ss}\) such that \(K_{{\mathbb {C}}}\cdot z=({{\mathbb {C}}}^d)^{\alpha -ss}\cap \overline{K_{{\mathbb {C}}}\cdot z}\) and such a z is unique up to a transformation of \(K_{{\mathbb {C}}}\).

\(\alpha \)-semistable points are characterized as follows:

Proposition 3

The following are equivalent for each \(z\in {{\mathbb {C}}}^d\):

  1. (1)

    z is \(\alpha \)-semistable;

  2. (2)

    \(\alpha \) satisfies the following:

    $$\begin{aligned} \alpha \in \sum _{j\in J_z}{{\mathbb {Q}}}_{\ge 0}(\iota ^*u^j/2\pi ), \end{aligned}$$

    where \(J_z\) denotes \(\{j\in \{1, \dots , d\}\mid z_j\ne 0\}\);

  3. (3)

    \(\alpha \) is in the cone generated by \((\iota ^*u^j/2\pi )_{j\in J_z}\):

    $$\begin{aligned} \alpha \in \sum _{j\in J_z}{{\mathbb {R}}}_{\ge 0}(\iota ^*u^j/2\pi ); \end{aligned}$$
  4. (4)

    For each \(v\in {{\mathbb {C}}}\backslash \{0\}\), \(\overline{K_{{\mathbb {C}}}\cdot (z, v)}\) does not intersect with \({{\mathbb {C}}}^d\times \{0\}\).

Proof

\((1)\Leftrightarrow (2)\) This can be proved by the same argument as in the proof of [23, Lemma 3.4].

\((2)\Leftrightarrow (3)\) This follows from the general theory of polyhedral convex cones. See [12].

\((1)\Rightarrow (4)\) Suppose z is \(\alpha \)-semistable. We take an \(f\in R_{n,\alpha }\) such that \(n\in {{\mathbb {Z}}}_{>0}\) and \(f(z)\ne 0\). We define a polynomial \({\hat{f}}(x, y)\) by \({\hat{f}}(x, y):=y^nf(x)\). Then we have the following:

$$\begin{aligned} {\hat{f}}(x, y)|_{\overline{K_{{\mathbb {C}}}\cdot (z, v)}}&\equiv v^nf(z), \\ {\hat{f}}(x, y)|_{{{\mathbb {C}}}^d\times \{0\}}&\equiv 0, \end{aligned}$$

and thus (4) holds.

\((4)\Rightarrow (1)\) Suppose (4) holds. Then there exists a polynomial \({\hat{f}}(x, y)\) such that

$$\begin{aligned} {\hat{f}}(x, y)|_{\overline{K_{{\mathbb {C}}}\cdot (z, v)}}&\equiv 1, \\ {\hat{f}}(x, y)|_{{{\mathbb {C}}}^d\times \{0\}}&\equiv 0. \end{aligned}$$

The polynomial \({\hat{f}}(x, y)\) can be written as \({\hat{f}}(x, y)=yf_1(x)+\cdots +y^mf_m(x)\). Take an \(n\in \{1, \dots , m\}\) such that \(f_n(x)\ne 0\). Then we have \(f_n\in R_{n,\alpha }\) and \(f_n(z)\ne 0\). \(\square \)

The closed orbits are characterized as follows:

Proposition 4

The following are equivalent for each \(z\in {{\mathbb {C}}}^d\):

  1. (1)

    z is \(\alpha \)-semistable and the \(K_{{\mathbb {C}}}\)-orbit is closed in \(({{\mathbb {C}}}^d)^{\alpha -ss}\):

    $$\begin{aligned} K_{{\mathbb {C}}}\cdot z=\overline{K_{{\mathbb {C}}}\cdot z}\cap ({{\mathbb {C}}}^d)^{\alpha -ss}; \end{aligned}$$
  2. (2)

    \(\alpha \) satisfies the following:

    $$\begin{aligned} \alpha \in \sum _{j\in J_z}{{\mathbb {Q}}}_{>0}(\iota ^*u^j/2\pi ); \end{aligned}$$
  3. (3)

    \(\alpha \) is in the interior of the cone generated by \((\iota ^*u^j/2\pi )_{j\in J_z}\):

    $$\begin{aligned} \alpha \in \sum _{j\in J_z}{{\mathbb {R}}}_{>0}(\iota ^*u^j/2\pi ); \end{aligned}$$
  4. (4)

    The following holds:

    $$\begin{aligned} \sum _{j\in J_z}{{\mathbb {R}}}(\iota ^*u^j/2\pi )=\sum _{j\in J_z}{{\mathbb {R}}}_{\ge 0}(\iota ^*u^j/2\pi )+{{\mathbb {R}}}_{\ge 0}(-\alpha ); \end{aligned}$$
  5. (5)

    For each \(v\in {{\mathbb {C}}}\backslash \{0\}\), \(K_{{\mathbb {C}}}\cdot (z, v)\) is closed;

  6. (6)

    The following holds:

    $$\begin{aligned} \mu _K^{-1}(-\alpha )\cap K_{{\mathbb {C}}}\cdot z\ne \emptyset . \end{aligned}$$

Proof

\((1)\Rightarrow (5)\) Suppose (1) holds. By the general theory of algebraic groups, there uniquely exists a closed orbit which is contained in \(\overline{K_{{\mathbb {C}}}\cdot (z,v)}\). Let \(K_{{\mathbb {C}}}\cdot (z^\prime , v)\) be such a closed orbit. Then by Proposition 3, \(z^\prime \in ({{\mathbb {C}}}^d)^{\alpha -ss}\). We take a sequence \((g_i)_{i\in {{\mathbb {N}}}}\) such that

$$\begin{aligned} (z^\prime , v)=\lim _{i\rightarrow \infty }g_i\cdot (z,v). \end{aligned}$$

Therefore we have \(z^\prime =\lim _{i\rightarrow \infty }g_i\cdot z\), and thus we see \(z^\prime \in \overline{K_{{\mathbb {C}}}\cdot z}\cap ({{\mathbb {C}}}^d)^{\alpha -ss}\). Then (5) holds.

\((5)\Rightarrow (1)\) Suppose (5) holds. Let \(z^\prime \in \overline{K_{{\mathbb {C}}}\cdot z}\backslash K_{{\mathbb {C}}}\cdot z\). We take a sequence \((g_i)_{i\in {{\mathbb {N}}}}\) so that \(z^\prime =\lim _{i\rightarrow \infty }g_i\cdot z\). Since \(K_{{\mathbb {C}}}\cdot (z,1)\) is closed, we have \(\lim _{i\rightarrow \infty }|\chi _\alpha (g_i)^{-1}|=\infty \). This implies that \(\lim _{i\rightarrow \infty }(g_i^{-1}z^\prime , \chi _{\alpha }(g_i))\in {{\mathbb {C}}}^d\times \{0\}\) and thus we have \(z^\prime \notin ({{\mathbb {C}}}^d)^{\alpha -ss}\).

\((2)\Leftrightarrow (3)\Leftrightarrow (4)\) This follows from the general theory of polyhedral convex cones. See [12].

\((3)\Leftrightarrow (6)\) We shall prove this in Proposition 5.

\((4)\Leftrightarrow (5)\) See [29, pp.30-31]. \(\square \)

The equivalence of (2) and (3) holds for any \(\lambda \in k^*\):

Proposition 5

Let \(\lambda \in k^*\) and \(z\in {{\mathbb {C}}}^d\). We define a functional \(l_{\lambda , z}:k\rightarrow {{\mathbb {R}}}\) by

$$\begin{aligned} l_{\lambda , z}(v):=\frac{1}{4}\sum _{j=1}^d|z_j|^2e^{2\langle u^j, v\rangle }-\langle \lambda , v\rangle . \end{aligned}$$

Then the following are equivalent:

  1. (1)

    \(\lambda \) is in the interior of the cone generated by \((\iota ^*u^j/2\pi )_{j\in J_z}\):

    $$\begin{aligned} \lambda \in \sum _{j\in J_z}{{\mathbb {R}}}_{>0}\iota ^*u^j; \end{aligned}$$
  2. (2)

    The following holds:

    $$\begin{aligned} \mu _K^{-1}(-\lambda )\cap K_{{\mathbb {C}}}\cdot z\ne \emptyset ; \end{aligned}$$
  3. (3)

    \(l_{\lambda , z}\) attains a minimum.

Moreover if v and \(v^\prime \) be minimizers of \(l_{\lambda , z}\), then \(v-v^\prime \) is in the orthogonal complement of \(\sum _{j\in J_z}{{\mathbb {R}}}\iota ^*u^j\).

Proof

We assume that \((\iota ^*u^j)_{j\in J_z}\) generates \(k^*\) for simplicity. Then a direct computation shows that \(l_{\lambda , z}\) is strictly convex. We also see that for each \(v\in k\), v is a critical point of \(l_{\lambda , z}\) if and only if the following holds.

$$\begin{aligned} \lambda =\frac{1}{2}\sum _{j=1}^de^{2\langle u^j, v\rangle }|z_j|^2\iota ^*u^j. \end{aligned}$$

Therefore (2) and (3) are equivalent. Clearly, (2) implies (1). Assume that (1) holds. We show that (3) holds. From the assumption, there exists a positive numbers \((s_j)_{j\in J_z}\) such that \(\lambda =\sum _{j\in J_z}s_j\iota ^*u^j.\) Then the functional \(l_{\lambda ,z}\) is denoted as

$$\begin{aligned} l_{\lambda ,z}(v)=\sum _{j\in J_z}(|z_j|^2e^{2\langle u^j,v\rangle }-s_j\langle u^j,v\rangle ). \end{aligned}$$

This implies that \(\lim _{t\rightarrow \infty }l_{\lambda ,z}(tv)=\infty \) for each \(v\ne 0\) and thus the functional \(l_{\lambda , z}\) attains a minimum. \(\square \)

From Proposition 2, Proposition 4 and Proposition 5, we have the following:

Corollary 3

The following map is bijective:

$$\begin{aligned} \mu _K^{-1}(-\alpha )/K\longrightarrow ({{\mathbb {C}}}^d)^{\alpha -ss}/\sim . \end{aligned}$$

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Miyatake, N. Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space. Geom Dedicata 214, 651–669 (2021). https://doi.org/10.1007/s10711-021-00632-z

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