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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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
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
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
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
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
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
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
Let \(R_\alpha \) be the invariant ring for the above action:
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
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:
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:
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)
z is \(\alpha \)-semistable;
-
(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)
\(\alpha \) 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)
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:
and thus (4) holds.
\((4)\Rightarrow (1)\) Suppose (4) holds. Then there exists a polynomial \({\hat{f}}(x, y)\) such that
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)
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)
\(\alpha \) satisfies the following:
$$\begin{aligned} \alpha \in \sum _{j\in J_z}{{\mathbb {Q}}}_{>0}(\iota ^*u^j/2\pi ); \end{aligned}$$ -
(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)
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)
For each \(v\in {{\mathbb {C}}}\backslash \{0\}\), \(K_{{\mathbb {C}}}\cdot (z, v)\) is closed;
-
(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
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
Then the following are equivalent:
-
(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)
The following holds:
$$\begin{aligned} \mu _K^{-1}(-\lambda )\cap K_{{\mathbb {C}}}\cdot z\ne \emptyset ; \end{aligned}$$ -
(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.
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
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:
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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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DOI: https://doi.org/10.1007/s10711-021-00632-z