Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options not deleted 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.

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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249
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 …
Bombyx mori's user avatar
  • 6,249