# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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### Is Kähler current class representable by semipositive forms?

A compact complex manifold is called in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a ...
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### A problem of the volume form of Kähler manifold in the paper of Yau's proof of Calabi conjecture

[This question arises from a look at the paper Shing-Tung Yau, "On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I", Comm. Pure Appl. Math., 31 (...
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### Upper bound on the bisectional curvature

This is a follow-up to the question Schwarz lemma and bisectional curvature lower bound. Looking at the same note Song and Weinkove - Lecture notes on the Kähler–Ricci flow, page 24, the first line ...
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### Inner product on global sections of positive line bundle

Let $\Sigma = S^2$ be thought of as a Riemann surface, and let $L$ be a Hermitian line bundle on $\Sigma$ with curvature $2$-form $-2 \pi i \Omega \in \Omega^2(\Sigma, \mathbb{R})$. Then $L$ is a ...
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### Characterize Hermitian-Einstein metric on $E$ using the tautological bundle $\mathcal{O}_E(1)$

Let $E\to X$ be a holomorphic vector bundle. Denote by $\mathbb{P}(E)\to X$ its projectivisation and $\mathcal{O}_E(1)\to \mathbb{P}(E)$ the associated tautological line bundle. I would like to know ...
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### Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...
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### Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?

The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
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### Fujiki class $\mathcal C$ with a symplectic structure

Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ ...
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Let $(X,\omega)$ be a Kahler manifold, denote by $\Lambda$ the dual of the Lefshetz operator $\omega\wedge$ (see e.g. Dual Lefschetz Operator and Contraction with the Fundamental Form). Let $\zeta\in\... 1 vote 0 answers 203 views ### Hironaka's construction for compact Kähler manifolds In Hartshorne's book 《Algebraic Geometry》 p.443, the author introduces a construction of a non-projective complex manifold from a projective one. His method can be summarized as following: Let$X$be ... 6 votes 0 answers 92 views ### Kahler property and finite covering Let$(M,\omega)$be a compact symplectic manifold and$\pi:\tilde M\to M$a finite covering. Clearly$(\tilde M,\pi^*\omega)$is a compact symplectic manifold. Suppose we know that$(\tilde M,\pi^*\...
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In Hartshorne's book 《Algebraic geomery》 p.443, the author gives an explanation of Hironaka's example on non-Kähler deformation of compact Kähler manifolds, his construction can be summarised as ...
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### Dependence of the space of holomorphic 1-forms on the complex structure

I am looking for a reference for the following fact: Assume $M$ is a closed manifold admitting complex structures of Kahler type. Then the space of holomorphic 1-forms on $M$ with respect to a Kahler-...
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### Global sections appearing in Dolbeault complex with values in vector bundle

Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex ...
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### Yau proof of $K_X>0$ using a non-smooth metric which restricts to a metric of negative holomorphic sectional curvature on all curves

In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote: Negative holomorphic sectional curvature is a rather ...
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### Unobstructed deformations of Hamiltonian manifolds

It is well known that compact complex manifolds with $H^2(X,T_X)=0$ and compact Kähler manifolds with trivial canonical bundle have unobstructed deformations, but besides them, are there other ...
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### Does every $\bar\partial$ harmonic form being $\partial$ closed make a manifold Kähler?

I'm reading Tian's paper 《Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric》, in page 635, there is a statement that: For a compact Kähler ...
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Let $V$ be a rational Hodge structure of degree $k$. Precisely, $V$ is a finite dimensional $\mathbb{Q}$-vector space whose complexification admits a decomposition $V_\mathbb{C} = \oplus_{p+q=k} V^{p,... 4 votes 1 answer 195 views ### Example of a Kähler manifold with certain properties I am looking for compact Kähler manifolds of dimension$3$with the following 2 properties: 1.$c_1(K_X)=c[\omega],c>0$where$\omega$is the Kähler form on$X$. 2.$1+h^{0,3}+h^{1,1}=h^{0,1}$It's ... 1 vote 0 answers 131 views ### Decomposition of a$(1,1)$form Let$X$be a compact Kähler three-fold and$\phi$be a Harmonic$(0,2)$-form, then$*(\phi\wedge\bar\phi)$is a$(1,1)$form. Hence it can be written as$\bar\partial\alpha+\bar\partial^*\beta+H$for ... 1 vote 1 answer 176 views ### An analogue of the Poisson bracket in contact geometry? I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question: McDuff and Salamon define an analogue of the Poisson bracket ... 2 votes 1 answer 106 views ### Are the odd dimensional spheres Poisson homogeneous spaces? Are the odd dimensional spheres$S^{2n+1}$, for$n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld? 4 votes 0 answers 109 views ### do cohomologically Kähler classes extend to Kähler classes? Let$f: X \to S$be a proper morphism from a complex manifold to a small disc which is smooth away from$Y = f^{-1}(0)$, an snc divisor. A class$\omega \in H^2(Y)$is called cohomologically Kähler if ... 5 votes 1 answer 228 views ### Computing the invariants of ball quotient surfaces The two-dimensional complex unit ball$B$has group of biholomorphic automorphisms$PU(2,1)$. If$Γ$is an arithmetic subgroup of$PU(2,1)$, the quotient$Γ\text{\\}B$is an orbifold. Taking its ... 5 votes 1 answer 188 views ### Volume of singular Kahler metric Let$X$be a compact complex manifold of complex dimension$n$and let$\omega$be a smooth Kahler form on it. Let$Y \subset X$be a complex (possibly singular) hypersurface and let$u: X \setminus Y ...
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Two well-known slogans are A Sasakian manifold is the odd dimensional analogue of a Kähler manifold and A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold Does this ...
Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau \$(c_{1,\...