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Results tagged with complex-geometry
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user 13268
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.
2
votes
Complex manifolds with spanning sets of holomorphic tensor fields
A very incomplete answer: if $c_1<0$ then there are no global sections of these tensor bundles: Kobayashi, First Chern class and holomorphic tensor fields, Nagoya Math. J., vol. 77, 1980, theorem A.
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7
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Accepted
Complex manifolds with spanning sets of holomorphic vector fields
On any compact complex manifold, the set of all global holomorphic vector fields is a finite dimensional Lie algebra. They span the tangent space at every point just when each component of the manifol …
- 26.3k
3
votes
What is a meromorphic connection?
A meromorphic connection is a meromorphic section of the bundle whose local sections are local connections, sometimes called the connection bundle. In a local trivialization, a meromorphic connection …
- 26.3k
4
votes
Complex structure on product of two $n$-dimensional real manifolds
I assume you are asking if $M \times N$ admits a complex structure.
I believe that there is no result known on this question, although we might have something to say about the existence of an almost c …
- 26.3k
7
votes
Analogue of Infinitesimal Schwarzian for holomorphic $(G,X)$-manifolds
Take a (holomorphic) Cartan connection $E \to M$ modelled on a (complex) homogeneous space $(X,G)$, say $X=G/H$. Let $\omega$ be the Cartan connection on $E$. Every (holomorphic) vector field $v$ on $ …
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2
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A question on real surfaces on K3 surfaces.
In local holomorphic coordinates $(z,w)=(x,y,u,v)$, write the 2-form as $\omega=f(z,w) dz \wedge dw$. (I assume you meant a holomorphic 2-form.) If we have a real 2-plane $P \subset T_x X$, and if $dx …
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4
votes
Domain of Holomorphy
Quoting Steven Krantz, Function Theory of Several Complex Variables, p. 6: the definition of domain of holomorphy is complicated because we must allow for the possibility (when dealing with an arbitra …
- 26.3k
2
votes
Accepted
Real diffeomeorphism preserving the space of Holomorphic vector fields
Take complex vector space $V$, say of complex dimension $n$. Take a complex linear map $A \colon V \to V$ whose eigenvalues $\lambda$ all satisfy $|\lambda|>1$. The group generated by $A$ acts on $V-0 …
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4
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Why Calabi-Yau manifolds should be complex?
You might consider other real forms of $SU(n)$; a Riemannian manifold with holonomy in such a real form will then arise as extra dimensions in string theories with reduced holonomy and a parallel spin …
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2
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Closure of orbit in complex manifold
No. Take a holomorphic translation vector field on the complex torus, whose orbits are translates of the dense image of some generic complex 1-dimensional linear subspace. The flow is complete, by com …
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0
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Newlander-Nirenberg in dimension 2
As Mohan Ramachandran mentioned in a comment above, there is a short and clear proof in Adrien Douady and X. Buff, Le théorème d’intégrabilité des structures presque complexes, The Mandelbrot set, the …
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1
vote
Accepted
Hodge numbers of compact Ricci-flat Kaehler manifold
After replacing by a finite covering space, you must have $h^{n0}=1$, because you have a parallel volume form, so nowhere zero, and any other holomorphic volume form is a holomorphic multiple, so a co …
- 26.3k
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Regarding definition of Kobayashi length
The vector $\xi$ is tangent to the complex line spanned by $\xi$. The significance of "infinitesimal" is that this definition defines a length of a tangent vector (in the sense of differential geometr …
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6
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Accepted
Holomorphic extension of an action by a compact Lie group on a complex homogeneous manifold
I think this should work for a connected group $G$. The Lie algebra action extends, by multiplying by $J$, to a complex Lie algebra action. By compactness of $M$, these vector fields are all complete. …
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0
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Kähler metric on a Zariski open subset of a non-Kähler manifold
You might be interested in the theory of complex manifolds of class C, due to Fujiki: a complex manifold is of class C if it is bimeromorphic to a compact Kaehler manifold. The concept was introduced …
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