# Tagged Questions

A complex manifold with a Hermitian metric is called a Kähler manifold if the (1,1) form that gives its Hermitian metric is a closed differential form.

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### Holomorphic vector bundles on almost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
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### Ricci curvature of sum of metrics

Is there an estimate for Ric$(g+h)$ in terms of Ric$(g)$ and Ric$(h)$, where $g,h$ are smooth Riemannian metrics? More specifically can one say that the eigenvalues will decrease (resp. increase) if ...
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### nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
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### On Dolbeault cohomology and Dolbeault operator

I'm trying to construct ladder operators on cohomology space, I searched for a similar procedure but I can't find anything. To be clearer, I consider the cohomology space of a compact Kähler manifold ...
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### Is this computation of the Christoffel coefficients on a Kähler manifold correct?

Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of ...
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### What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
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### What is a pseudo-Kähler manifold?

I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that ...
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### Computing real de Rham cohomology of Hironaka's 3-manifold example

I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two ...
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### Clarification about the definition of Calabi-Yau manifold

There are a lot of different definitions of a Calabi-Yau manifold. Roughly, we can divide them in two sets, see Wikipedia https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold . I will refer to ...
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### complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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### Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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### Intuition for Kähler manifolds?

Define a Kähler manifold to be a complex manifold whose associated (1,1) form is closed. One can show this condition leads to many interesting properties. For example, the Hodge and Lefschetz ...
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### Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...