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Results tagged with complex-geometry
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user 124749
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.
5
votes
1
answer
300
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Vanishing theorems on a non-compact manifold
In complex geometry, various vanishing theorems for cohomology
groups of a hermitian line bundle E over a compact complex manifold X have been found.
My question is
Is there some vanishing theore …
- 391
2
votes
0
answers
119
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Is there Hodge isomorphism between Dolbeault and Harmonic on noncompact manifold
As is well known , Hodge theorem tells us
Let $(X, g)$ be a compact hermitian manifold. Then the canoni.
cal projection $\mathcal{H}_{\bar{\partial}}^{p, q}(X, g) \rightarrow H^{p, q}(X)$ is an isomo …
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1
vote
0
answers
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Are positivity for forms and that for currents consistent when talking about smooth forms?
Let $X$ be a complex manifold and $\theta$ a smooth $(1,1)$-form on $X$.
(1) If $\theta>0$ in the sense of currents, then can we deduce that $\theta>0$ in the sense of forms?
(2) If $\theta>0$ in the …
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2
votes
1
answer
184
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Relations between Dolbeault cohomology and the corresponding $L^2$-cohomology
Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p, …
- 391
4
votes
1
answer
309
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Existence of plurisubharmonic functions on complex manifolds
Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets.
Conjecture: There must be strictly plurisubharmonic functions on $X$ .
Is it true?
- 391
1
vote
0
answers
132
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Essential Steinness of projective manifold
As we all know, a projective manifold is an essentially Stein manifold. Here, we use the definition as follows: A Kähler manifold Y is said to be essentially Stein if there exists an analytic hypersur …
- 391
2
votes
0
answers
96
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What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorp...
What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle or other holomorphic vector bundle over a complex manifold ?Do we have anything similar wit …
- 391
1
vote
0
answers
152
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Lebesgue dominated convergence for currents
I am learning current theory presently. In Chap 3.3-Definition of Monge-Ampère Operators, J.-P. Demailly, Complex analytic and differential geometry, I am a little confused as follows.
Let $X$ be a …
- 391
7
votes
0
answers
192
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The comparison between Dolbeault cohomology and $L^2$ cohomology
Let $X$ be a complex manifold. Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact c …
- 391
2
votes
0
answers
71
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About the current of finite mass
In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ i …
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