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Results tagged with complex-geometry
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user 4054
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.
16
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Wanted: an example of a natural non-Kähler metric on a Kähler manifold
Let $X$ be a Kähler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kähler form of $h$. One of several equivalent conditions …
- 499
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votes
1
answer
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Kleiman criterion for Kähler classes
Demailly and Paun proved the following characterization of nef classes on a compact Kahler manifold:
Theorem 18.13(a). Let $X$ be a compact Kähler manifold. A $(1,1)$-class $\alpha$ on $X$ is nef if a …
- 6,871
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votes
0
answers
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A paper that proves the blowup of the projective plane has positive holomorphic sectional cu...
I'm convinced I've read a paper where the authors prove that the blowup of the projective plane in a single point admits a metric of positive holomorphic sectional curvature. This was not the main foc …
- 4,343
5
votes
1
answer
570
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Translation of Kähler's "Über eine bemerkenswerte Hermitesche Metrik"
Has anyone translated Erich Kähler's "Über eine bemerkenswerte Hermitesche Metrik" into English or French? (Preferably, but I'll take anything.)
- 4,713
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2
answers
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What is the holomorphic sectional curvature?
Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of …
CommunityBot
- 1
6
votes
0
answers
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The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes
Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an alge …
- 4,343
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answers
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References for holomorphic foliations
I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds.
Any little helps, but I'm particularily interested in problems of the type where we have a hermitian man …
- 366
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2
answers
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Is the deformation limit of Ricci-flat Kahler manifolds Kahler?
Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are compac …
CommunityBot
- 1
8
votes
2
answers
556
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Inequality on Kähler classes
Let $X$ be a compact Kähler manifold of complex dimension $n$, and let
$\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz
operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. Th …
CommunityBot
- 1
4
votes
0
answers
472
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Is there any advantage to knowing that Gauss-Manin is Hermitian flat?
Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with cu …
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votes
2
answers
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Which almost complex manifolds admit a complex structure?
I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau …
CommunityBot
- 1
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votes
1
answer
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Calculating a second fundamental form in the space of hermitian metrics
Let $X$ be a compact Kahler manifold and let $\mathcal M$ denote the space of hermitian metrics on $X$. We'll identify a hermitian metric with a smooth, real and positive $(1,1)$-form $\omega$. Let $\ …
CommunityBot
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2
answers
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Is a holomorphic family whose fibers are all smooth locally trivial?
Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers $X_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : …
- 12.4k
4
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answers
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Ramification divisor and degenerate locus of jacobian
Let $f : X \to Y$ be a finite morphism between compact complex manifolds of the same dimension $n$. We denote by $R_f \subset X$ the ramification divisor of $f$ and by $J_f \subset X$ the set of point …
- 149k
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answers
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"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X …
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