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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.
4
votes
1
answer
426
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Example of a variety with explicit cohomology ring and Kahler cone
I'm looking for some fairly explicit varieties to use as (counter?-)examples for my thesis and I'd appreciate any suggestions. I need a smooth projective variety $X$ of general type that satisfies:
…
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9
votes
1
answer
1k
views
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 $\ …
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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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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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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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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
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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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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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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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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4
votes
1
answer
575
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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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16
votes
4
answers
1k
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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 …
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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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1
vote
0
answers
717
views
Does the tangent bundle of this fiber product split?
Let $\mathcal X \to S$ be the local universal family of an elliptic curve, and let $E \to S$ be a vector bundle over $S$. Then we can form the fiber product $\mathcal Y = \mathcal X \times_S E$, which …
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6
votes
1
answer
1k
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Harmonic forms on Ricci-flat Kahler manifolds
Let $X$ be a compact Kahler manifold with $c_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian.
…
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