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Results tagged with complex-geometry
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user 392184
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
12
votes
2
answers
723
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Non-Kähler pseudo-Kähler manifolds
A pseudo-Kähler manifold is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a …
- 473
9
votes
0
answers
379
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Kähler metric on the Hilbert scheme of points on a surface
Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler m …
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7
votes
1
answer
747
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What is the definition of a Calabi-Yau metric on a non-compact manifold?
There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:
[1] Li, Y. A new complete Calabi-Yau metric on …
- 473
6
votes
1
answer
207
views
Existence of a real structure on the tangent bundle of a complex manifold
Let $X$ be a compact complex manifold. What are obstructions to the existence of a real subbundle $V$ of $TX$ such that $TX = V \otimes \mathbb{C}$?
For example, does $\mathbb{CP}^n$ have such a struc …
- 473
4
votes
0
answers
120
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Does unobstructedness depend on the representative of a Kodaira-Spencer class?
Let $X$ be a compact complex manifold. One says that a Kodaira-Spencer class $c \in H^1(X, TX)$ is unobstructed if there exist $\phi_n \in \Omega_X^{0,1}(T^{1,0}X)$ for $n \ge 1$ such that $c = [\phi_ …
- 473
4
votes
1
answer
391
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Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles
Question. What are examples of compact complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles?
A family of examples are, of course, holomorphically symplectic m …
- 473
3
votes
1
answer
355
views
Extension of closed $(1, 1)$-forms
Let $X$ be a compact Kähler manifold and $S \subset X$ a closed complex submanifold. Given a closed $(1, 1)$-form $\alpha$ on $S$, is there always a closed $(1, 1)$-form $\beta$ on a neighborhood of $ …
- 473