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Results tagged with complex-geometry
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user 3709
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.
20
votes
5
answers
7k
views
Griffiths and Harris reference
Trying to read the section on Poincare duality from Griffiths and Harris is a nightmare. I want to know if there is a place where Poincare duality and intersection theory are done cleanly and rigorous …
- 3,383
12
votes
2
answers
541
views
Relationship between the Radon transform and Twistor spaces
I have often heard that the theory of Twistor spaces is ``a complex analogue" of the Radon transform. What is the precise connection ?
- 3,383
11
votes
What is a Futaki invariant, what is the intuition behind it, and why is it important?
The Futaki invariant $F(X,[\omega])$ is a quantity that needs two pieces of information on a compact complex manifold $M$.
1) A Kahler class $[\omega]$
2) A holomorphic vector field $X$.
It is an …
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11
votes
2
answers
3k
views
$\partial \bar{\partial}$ lemma for contractible domains
Question. Is every $(p, \, p)$ closed form ($p\geq1$) in a contractible open set of $\mathbb{C}^n$ $\partial \bar{\partial}$ exact?
We know that every $d$-exact $(p, \,p)$-form on a compact Kahl …
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10
votes
0
answers
342
views
Local meaning of the Pfaffian of the curvature
The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can p …
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10
votes
1
answer
788
views
Vector bundles on Stein manifolds
This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold $X$ to $\operatorname{Gr}(k,n)$ (the Grassmannian of $k$ planes in $\ …
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7
votes
5
answers
3k
views
An example of a complex manifold without a finite open cover
Are there non-compact complex manifolds that
a) Don't embed in C^n (holomorphically)
and
b) Cannot be covered by a finite number of coordinate open sets?
If b) can be satisfied, then I think so can a) …
- 3,383
6
votes
0
answers
428
views
A non-elliptic PDE
I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X :
$\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial …
- 3,383
6
votes
1
answer
1k
views
Chern classes generating cohomology
The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my questio …
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5
votes
1
answer
212
views
$L^p$ stability of the Beltrami equation
Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} …
- 3,383
5
votes
1
answer
1k
views
Quillen metric definition
I am confused as to why a regularised determinant is used in the definition of Quillen's metric on the determinant line bundle defined over the space of $\bar{\partial}$ operators on a (hermitian) vec …
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4
votes
1
answer
215
views
Examples of surfaces with negative Kahler curvature operator
Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces …
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4
votes
2
answers
2k
views
Branched covers of compact Riemann surfaces
Let $S$ be a compact R.S of genus $\geq 2$. In the paper "Stable and unitary vector bundles on compact Riemann surfaces" (by Narasimhan and Seshadri), they claim that there is a branched covering map …
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4
votes
0
answers
189
views
Chern-Weil theory for coherent subsheaves
If $(E,h)$ is a holomorphic vector bundle over a compact complex manifold $M$ (which is a projective variety), and $S$ is a coherent subsheaf of $M$, then $S$ is secretly a holomorphic vector bundle o …
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4
votes
the central issues in complex geometry
There also problems along the lines of proving L2 extension theorems for vector valued forms (there are such theorems already like the Ohsawa-Takegoshi extension theorem however, that applies to (n,1) …
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