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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.
5
votes
1
answer
1k
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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
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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1
vote
0
answers
140
views
Equivariant Kodaira embedding
Suppose $X$ is a compact complex manifold acted upon by biholomorphisms by a complex Lie group. Suppose $L$ is an equivariant line bundle which is also ample (in the sense that it admits an equivarian …
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2
votes
0
answers
166
views
Why only the first two Chern classes in the BMY and KL inequalities?
The Bogomolov-Miyaoka-Yau inequality for compact complex manifolds with ample canonical bundle and the Kobayashi-Lubke inequality for holomorphic stable vector bundles involve the first two Chern clas …
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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 …
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7
votes
5
answers
3k
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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) …
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4
votes
0
answers
782
views
Atiyah-Bott Yang-Mills connections
In Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces, a special case of what they do is to prove that Unitary Yang-Mills connections over a R.S $M$ are in bijective correspondence with u …
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1
vote
1
answer
383
views
Holomorphic h-principle for compact manifolds
The Oka principle for Stein manifolds says (roughly) that the only obstructions for "things" are topological obstructions (for instance every smooth complex vector bundle admits a holomorphic structur …
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2
votes
1
answer
217
views
Number of semistable subbundles of a semistable bundle
Is the following true ? If so, is there a quick proof of it ? (Perhaps using the uniqueness of the graded object associated to a Jordan-Holder filtration or maybe otherwise)
Suppose $E$ is an $\omega …
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1
vote
0
answers
123
views
Is every curve on a projective three-fold a homology-theoretic complete intersection of sorts?
Let $C$ be a curve on a smooth projective three-fold $M$ equipped with the restriction of the Fubini-Study metric $\omega$. I'd like to know if there exists a surface $S$ such that for every closed $( …
- 3,383
6
votes
1
answer
1k
views
Chern classes generating cohomology
The fact that Chern classes are Hodge classes (and are rational combinations of algebraic cycles) is a part of the proof of the "Gauss Bonnet theorem" (as given in Griffiths and Harris). So my questio …
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20
votes
5
answers
7k
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Griffiths and Harris reference
Trying to read the section on Poincare duality from Griffiths and Harris is a nightmare. I want to know if there is a place where Poincare duality and intersection theory are done cleanly and rigorous …
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11
votes
2
answers
3k
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$\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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3
votes
1
answer
275
views
A (non-Kahler) metric on projectivised vector bundles
Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : Fir …
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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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