# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following: If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
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Let $M$ be a Kähler manifold and $V$ a singular hypersurface of $M$. Assume we obtain an embedded resolution $M^{\prime}$ of $V$ in $M$ by finitely many blow-ups along smooth centers. My question ...
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### prove a bondle is an indefinite Hermitian manifold which is Kahler if and only if the manifold is locally flat

Let $M(J,g)$ be an indefinite Kahler manifold, then $% TM(J^{H},g^{D})$ is an indefinite Hermitian manifold which is Kahler if and only if $M$ is locally flat. Here $J^{H}$ denotes the horizontallift ...
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### proof of the following theorm on simply-connected, complete indefinite Kahler manifold

can anyone help me prove the following therorm If $c\in \mathrm{I\!R}$ every connected, simply-connected, complete indefinite Kahler manifold of complex dimension $n$, of index 2s and of constant ...
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### Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
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### Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
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### $S^3$ as a Sasakian Manifold

Reading about Sasakian manifolds one come across two slogans: A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold." B) "A Sasakian manifold sits between two Kahler manifolds -...
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### Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler

Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example? (*)Under the assumption ...
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### Applications of Hodge-Riemann bilinear relations [closed]

I am wondering if the Hodge-Riemann bilinear relations have any further applications/ developments in Kahler or algebraic geometry. Let me briefly remind the statement. Given a compact Kahler ...
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### Control the convex combination of two classes on the boundary of the kahler cone

Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone ...
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### Vector bundle over compact complex manifold which is not holomorphic?

A vector bundle over a complex manifold is said to be holomorphic if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex ...
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### Projective embeddings of quotients of normal varieties

Let $X$ be a normal complex projective variety of dimension $m$, $G$ be a finite subgroup of $\mathrm{Aut}(X)$, and $Y = X / G$ be the quotient. I am particularly interested in the case where $X$ is a ...
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### Equality of the derivative of the exponential map on Kähler manifolds

Let $M$ be a Kähler manifold, $\omega$ its Kähler form and $J$ the complex structure. Moreover, let $V$ be a smooth (or even analytic, I am not sure if this is important) vector field on $M$, $p \in M$...
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### Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?

Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$
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### Formality and symplectic forms on a smooth manifold

I saw one paper which asks this question. "Let $(M,\omega,J)$ be a Kähler manifold. Then does $M$ admit a symplectic structure $\sigma$ of non-hard Lefschetz type?". I was wondering whether I could ...
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### Do non-compact Fano manifolds exist?

Suppose $(M,g, \omega)$ is a Kähler manifold with $\text{Ric}(g) = g$, i.e., $M$ is a Fano manifold. Is $M$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a ...
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### Type control of differential forms on total space of a family $X\to \Delta$

Let $\pi:X\to \Delta$ be a smooth family of compact Kahler manifolds over a small disk in $\mathbb C$. Let $\omega\in \mathcal{A}^{p,q}_X$ be a smooth exact form of type $(p,q)$ on the total space. I ...
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### Sectional curvature in complex manifold

Let $(X, \omega)$ be a Hermitian manifold .Say that the sectional curvature of X is negative is the same to say that the sectional curvature of the Hermitian metric $\omega$ is negative, otherwise, ...
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### Equivariant cohomology of $\text{Diff}S^1/ S^1$ and Virasoro

Consider $$\mathcal{M}\ =\ \text{Diff}S^1/S^1$$ which is a contractible complex manifold with an action of $\text{Diff}S^1$ by translations. It is claimed in page 358 of  that $\mathcal{M}$ has ...
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### Is there a Kähler manifold with no anti-holomophic involution?

That is, is there a Kähler manifold $X$ on which there is no map $$\tau:X\to X$$ such that $$d\tau\circ I=-I\circ d\tau$$ and $$\tau\circ \tau=\mathrm{Id}_X?$$
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### Unitary representations of Kähler groups deformable to one another as complex representations

Let $G=\pi_1(M)$ be the fundamental group of a compact Kähler manifold. Let $n$ be a non-negative integer. Then the set of homomorphisms $G\to GL(n, \mathbb{C})$ can be considered as a real variety $X$...
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### Positive-dimensional Seiberg-Witten moduli spaces

I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$). Of course, the result/conjecture is that ...
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### Automorphism group of compact hyperkähler manifolds

Let $M$ be a compact simply-connected hyperkähler manifold, and let $$\mathrm{Aut}(M)$$ be the automorphism group of $M$, i.e. the group of tri-holomorphic diffeomorphisms preserving the metric. ...
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### Kaehler analogue of very ample line bundle

In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a ...
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### Definition of Canonically polarized manifold?

Does anyone have a reference for the definition of a canonically polarized manifold? Typically, at least from what I have seen, a polarized manifold is a compact Kähler manifold $X$ together with an ...
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### Geometric intuition of $\inf_{y \in Y} R_{i\overline{i} j \overline{j}}$ on a compact Calabi--Yau manifold

Let $(X, \omega_X)$ be a compact Calabi--Yau manifold with Ricci-flat metric $\omega_X$. Let $R_{i \overline{j}k \overline{\ell}}$ denote the Riemannian curvature tensor of $(X, \omega_X)$. It is ...
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### First Chern class with sign

Let $(M,\omega)$ be a compact Kähler manifold with Kähler form $\omega$. Furthermore, denote by $c_{1}$ the first Chern class of $M$. Assume one of the following $c_{1}>0$, $c_{1}<0$ or $c_{1}=0$...
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### Non-compact hard Lefschetz theorem

For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See here or here for an overview of the result. What happens in the non-...
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### Kähler manifold with even-only singular cohomology

Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...
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### The effect of the Hodge $\star$ operator on the symplectic structure of a Kahler $4$ manifold

Let $(M,\omega, J, g)$ be a $4$ dimensional Kahler manifold. Put $\omega'=\star \omega$ where $\star$ is the Hodge operator associated the metric $g$. Is $(M,\omega ')$ a symplectic manifold? Is it ...
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### Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein

Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein? I was told that we can use the following method: Step ...
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### Kaehler varieties

Let $X\rightarrow D$ be a proper holomorphic map of complex-analytic spaces that is a submersion away from the origin. Suppose that the central fiber is the analytification of a reduced scheme ...
I am trying to make sense of this blog post. Let $D$ be the unit disk endowed with its standard complex structure. A family of complex-analytic spaces over a disk is a proper holomorphic map \$X\...