Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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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 ...
Jake Wetlock's user avatar
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2 votes
1 answer
124 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?
Jake Wetlock's user avatar
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4 votes
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116 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 ...
Dima Sustretov's user avatar
5 votes
1 answer
312 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 ...
LeechLattice's user avatar
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5 votes
1 answer
225 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 ...
complex's user avatar
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1 vote
1 answer
136 views

Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

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 ...
Dick Johnson's user avatar
6 votes
1 answer
786 views

The period map and the Kodaira--Spencer map

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,\...
GradStudent's user avatar
5 votes
1 answer
270 views

Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds

Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a ...
Jesse Madnick's user avatar
2 votes
1 answer
263 views

Examples of complex manifolds for which the logarithmic cotangent bundle is big, but the cotangent bundle is not big

Let $(X,D)$ be a log pair, with $X$ a projective manifold (or quasi-projective) and $D$ a divisor with simple normal crossings. I'd like to construct an example, or be pointed to a reference, for an ...
GradStudent's user avatar
8 votes
1 answer
322 views

An integration identity on $\mathbb{P}^{n-1}$

Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's ...
GradStudent's user avatar
1 vote
1 answer
362 views

Are smooth Schubert varieties Kähler? [closed]

Schubert variety $V$ is a special type of (possibly singular) subvarieties of a Grassmannian. Since the Grassmannians are Kähler manifolds (in fact projective varieties) are we able to conclude that ...
Boris Henriques's user avatar
3 votes
0 answers
145 views

Why does the bisectional curvature blow up?

Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, ...
Shiyu's user avatar
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2 votes
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175 views

A $\partial\bar\partial$ type problem in Kähler Geometry

On any compact Kähler manifold $M^n$ one can ask: given a closed $p,q$ form $\alpha$ on $M$ does $\exists\beta\in\Omega^{p-1,q-1}(M;\mathbb{C})$ such that $\alpha=\partial\bar\partial\beta.$ I am ...
Partha's user avatar
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21 votes
1 answer
1k views

Does every group arise as the fundamental group of a complete Kähler manifold?

The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
Michael Albanese's user avatar
2 votes
0 answers
109 views

The projectivity of a Kähler fiber space over a projective manifold

Let $X$ be a compact Kähler manifold, $B$ being a complex projective manifold, for a smooth fibration:$\pi:X\rightarrow B$ such that all the fibers are a projective manifold $Y$ with $H^i(Y,\mathcal O)...
Tom's user avatar
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14 votes
1 answer
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What makes a Kähler manifold projective?

Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\...
Tom's user avatar
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4 votes
1 answer
752 views

A contradiction caused by the Kähler identity and the formal adjoint relation

I found a contradiction in the Principle of Algebraic Geometry by G&H, section 1.2. I have post this on MSE but it didn't get enough attention. I couldn't sleep or eat or do anything else due to ...
XT Chen's user avatar
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2 votes
1 answer
461 views

Integration by parts on a Kähler manifold

I am trying to make sense of integration by parts on a Kähler manifold $X$ equipped with a Kähler metric $\omega$. Given two smooth real functions $f$ and $h$ on $X$, I want to write down the ...
penny's user avatar
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2 votes
0 answers
114 views

Metric of negative holomorphic sectional curvature

Let $X$ be a Kähler manifold which admits a Hermitian metric of negative holomorphic sectional curvature. Does $X$ admit a Kähler metric with negative holomorphic sectional curvature? This question is ...
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1 vote
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142 views

Kähler fiber space with base and fiber projective

Let $X$ be a Kähler manifold, $Y$ be a projective manifold, if $X$ exits a smooth fibration over $Y$ such that all the fibers are projective manifolds, then is $X$ a projective mannifold? If we do not ...
Tom's user avatar
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2 votes
0 answers
107 views

A tri-grading on the de Rham complex of a Lie group?

The tangent space of a (compact) Lie group $G$ is given by its Lie algebra. Assuming for convenience that $G$ is connected, its Lie algebra $\frak{g}$ decomposes into a Cartan part, as well as ...
Dick Johnson's user avatar
3 votes
0 answers
206 views

Is there a compact complex surface $X$ with $c_2(X)=7+6n$ and $c_1^2(X)=17+18n$?

As stated in [1], most pairs of positive integers $c_1^2$, $c_2$ satisfying $c_1^2+c_2=0$ $\mod 12$, the BMY inequality and the Noether inequality are actually Chern numbers of compact complex ...
LeechLattice's user avatar
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4 votes
0 answers
86 views

Curvature universal abelian variety

I am reading N.Mok's paper "Aspects of Kähler Geometry on Arithmetic varieties", I am especially interested in the computation of the curvature for the space $\mathcal{H}_g \times \mathbb{C}^...
user141601's user avatar
0 votes
1 answer
216 views

Is it true that a projective Kähler manifold of general type has a smooth canonical model and has no singular fibers?

A projective Kahler manifold $X$ of general type is a manifold which is projective and whose canonical bundle is big and nef. Let $\Phi: X \to X_{can}$ denote the map from $X$ to its canonical model. ...
Kaylynn's user avatar
6 votes
1 answer
330 views

Atiyah-Singer for Riemannian and Kaehler manifolds

I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
Dick Johnson's user avatar
7 votes
0 answers
267 views

Triviality of holomorphic vector bundles over $\mathbb{C}$

Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle. I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
Alessio Di Prisa's user avatar
4 votes
0 answers
213 views

Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric. I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
gigi's user avatar
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3 votes
0 answers
156 views

Ricci curvature of a Kahler current

Let $M$ be a compact Kahler manifold, with a divisor $D$, $\mathcal{H}_{\omega} = \{\varphi \in C^{\infty}(M - D) \cap C^{0}(M) : \omega_{\varphi} = \omega + \sqrt{-1} \partial \bar \partial \varphi &...
Chenghui Shan's user avatar
49 votes
4 answers
8k views

What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery ...
3 votes
1 answer
247 views

Curvature of varieties of log general type

Let $X$ be a projective manifold and $\Delta$ a divisor with simple normal crossings. Consider $X$ as the compactification of a quasi-projective variety $X_0$ with boundary $\Delta$, i.e. $X_0 = X \...
AmorFati's user avatar
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2 votes
1 answer
169 views

Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$

What is the Kahler cone of $\mathbb{C}P^1 \times \mathbb{C}P^n$ blown-up along a co-dimension two subvariety of the form $pt \times H$ where $H \subset \mathbb{C}P^n$ is a hyperplane?
mathdonkey's user avatar
2 votes
1 answer
430 views

About an explicit formula of the curvature tensor by holomorphic sectional curvatures

Let $(M, g)$ be a Riemannian manifold. Define the curvature tensor convention as follows. $$ R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$$ $$ R(X,Y,Z,W) = g(R(X,Y)Z, W)$$ ...
Yongmin Park's user avatar
5 votes
0 answers
182 views

Proof of Tian's constant

Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
Linda Lee's user avatar
4 votes
2 answers
547 views

Reading material for an analytical aspect of Kähler Geometry

This question was originally posted on MSE. But I would like to post it here to see whether anyone could recommend some reference for me. I am currently reading the paper "Three-circle theorem ...
ldgo's user avatar
  • 97
4 votes
0 answers
203 views

Rigid non-algebraic manifolds

The famous Kodaira problem asks: whether a compact Kähler manifold can always be deformed to a projective manifold? In order to provide a counterexample, one way is trying to construct a rigid compact ...
Tom's user avatar
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5 votes
1 answer
328 views

Fibrations in complex geometry

Let $X^n$ be a compact Kähler manifold with $K_X$ semi-ample, i.e., a sufficiently high power of $K_X$ is basepoint free. The associated pluricanonical system $| K_X^{\ell} |$ furnishes a birational ...
AmorFati's user avatar
  • 1,349
2 votes
0 answers
92 views

Regularity of a singular Kaehler Einstein metric

On a manifold $X$ of general type i.e. $X$ is projective and $c_1(K_{X})$ semiample. One can construct a singular Kaehler Einstein metric $\omega_{\infty}$ in $-c_1(X)$. In particular, $\omega_{\infty}...
ranzhang's user avatar
7 votes
1 answer
588 views

$H^{p,q}(X)$ versus $H^{q}(X, \bigwedge^p TX)$

Let $X$ be a Kahler manifold. To $X$ one can associate the cohomology groups $H^{p,q}(X)$, and $H^{(0,q)}(X, \bigwedge^p TX)$ with $TX$ being the holomorphic tangent bundle of $X$. Is there a general ...
dayar's user avatar
  • 319
3 votes
1 answer
193 views

Étale covers pulling back a very ample class to any integer multiple

Let $V$ be a smooth complex projective variety. Choose a very ample class $H\in H^2(V, \mathbb{Q})$. Can there exist finite étale morphisms $\phi_k:V\to V$ for each $k\geq 1$ such that $\phi^*_kH=kH$?
user avatar
6 votes
0 answers
152 views

Completion/Compactification of a Kähler metric on $\mathbb C^2$

Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$ \omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right), $$ where $\mu$ is a positive real ...
Robbixmaths's user avatar
1 vote
0 answers
930 views

Explicit construction of Fubini Study Metric

I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$ from Notes on canonical Kähler metrics on page 8 is remarked (Example 2.12 4.): Fix a Hermitian innerproduct on $\mathbb{C}^{...
user267839's user avatar
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1 vote
1 answer
213 views

Fixed locus of a Kahler $S^1$-action

Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold. It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
Filip's user avatar
  • 1,617
9 votes
1 answer
311 views

Complex structures on Hermitian symmetric space

Let $(M_1,g_1,J_1)$ and $(M_2,g_2,J_2)$ be two simply-connected Hermitian symmetric spaces, which are isometric as two Riemannian manifolds. Can we find an isometry $\varphi:M_1 \to M_2$ such that $$ \...
Totoro's user avatar
  • 2,515
3 votes
1 answer
194 views

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 ...
MOULI Kawther's user avatar
0 votes
0 answers
41 views

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 ...
MOULI Kawther's user avatar
5 votes
1 answer
403 views

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" ...
BinAcker's user avatar
  • 767
1 vote
1 answer
208 views

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 ...
Fofi Konstantopoulou's user avatar
2 votes
1 answer
195 views

$S^3$ 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 -...
Fofi Konstantopoulou's user avatar
7 votes
0 answers
286 views

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 ...
Eder Moraes's user avatar
3 votes
1 answer
1k views

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 ...
asv's user avatar
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