# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

422 questions
Filter by
Sorted by
Tagged with
157 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 ...
127 views

116 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 ...
161 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 ...
245 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 ...
267 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 ...
129 views

59 views

### example of noncompact kahler manifold which is not modification of Stein manifold

It is obvious that every Stein manifold is Kahler. But the gap between these two can be quite huge. If the Kahler manifold contains compact complex submanifolds of positive dimension, it cannot be ...
587 views

147 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. ...
279 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 ...
216 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 ...
126 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 ...
107 views

103 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?
224 views

514 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 ...
110 views

### Automorphism groups of Kähler–Einstein manifolds

Let $(X, \omega)$ be a compact Kähler manifold. We will say that $X$ is Calabi–Yau if the first Chern class of the anti-canonical bundle is trivial, in symbols: $c_1(-K_X)=0$; we will say $X$ is of ...
183 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 $\phi^*_kH=kH$?
### 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 ...