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Results tagged with complex-geometry
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user 18850
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.
6
votes
Theta functions on an elliptic curve and Serre duality
Here is a 'low-brow' approach. One type of the result you are talking about has been written up implicitly in Lang's book Introduction to Arakelov theory. The case for cohomology of the elliptic curv …
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5
votes
Accepted
Is there an example to show the Hodge decomposition fails on non-compact case?
The kernel and cokernel of an operator on a non-compact manifold could have infinite dimension. So the classical Hodge decomposition no longer holds. For a simple example, consider the unit open inter …
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4
votes
0
answers
542
views
Is Serre duality related to Pontryagin duality?
I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator …
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3
votes
Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth F...
As Deane Yang pointed out, in general, an elliptic operator of order $s$ maps $H^{k}\rightarrow H^{k-s}$ for functions defined on $\mathbb{R}^{n}$. The pseudo-differential operator on a compact manifo …
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2
votes
Quillen metric definition
In a nutshell, I think knowing $|D_1-D_2|$ is very small in operator norm does not enable you to control the jump of the dimension of $\ker D_1$ to $\ker D_2$. All we have is estimates like
$$
|D_2 v …
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0
votes
intersection of holomorphic curve with hyperplane
I think the general answer would be unbounded unless you restrict $f$ to be some special class of entire functions. But the reason for this is trivial; namely you can approximate any continuous comple …
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9
votes
3
answers
938
views
Examples of compact complex manifolds for which the $dd^c$ lemma does not hold
The well known $dd^{c}$ lemma in complex geometry claimed that
Let $X$ be a compact Kähler manifold. Let $p,q\ge 1$. Let $\eta$ be a
$(p,q)$ form on $X$ and assume $\eta$ is $d$-exact. Then the …
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