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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.
3
votes
1
answer
223
views
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 …
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2
votes
0
answers
90
views
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 …
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6
votes
0
answers
157
views
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 …
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8
votes
2
answers
556
views
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 …
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4
votes
0
answers
472
views
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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18
votes
2
answers
4k
views
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 …
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3
votes
2
answers
727
views
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 : …
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4
votes
2
answers
768
views
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 …
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4
votes
1
answer
228
views
Existence of nodal curves in a linear system
Let $S$ be a projective surface and $L$ an ample line bundle on $S$. The Severi variety $\mathcal V_{\mathcal L,\delta}$ parametrizes curves with $\delta$ nodes and no other singularities in the linea …
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12
votes
1
answer
540
views
Which complex manifolds embed into tori?
If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to em …
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5
votes
2
answers
302
views
Is the cup product of holomorphic $n$-forms with a fixed class injective?
Let $X$ be a compact Kahler manifold of complex dimension $n$. Fix a nonzero class $u \in H^1(X,T_X)$. This gives a linear morphism
$$
\phi_u : H^0(X,\Omega^n) \to H^{1}(X,\Omega^{n-1}),
\quad \sigma …
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6
votes
0
answers
223
views
Complex manifold with non-finitely generated canonical ring
P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have t …
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4
votes
1
answer
575
views
Automorphism group of ruled surface
Let $C$ be an elliptic curve over the complex numbers. Consider a nontrivial extension
$$
0 \to \mathcal O_C \to E \to \mathcal O_C \to 0
$$
of rank 2 of the structure sheaf of $C$. This defines a rul …
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5
votes
1
answer
570
views
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.)
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5
votes
1
answer
682
views
Tangent sheaf of a hom scheme
I apologize if this question is too basic, but I haven't been able to work this out for myself.
Let $X$ and $Y$ be projective schemes, say over the complex numbers. There exists a scheme $Hom(X,Y)$ p …
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