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Results tagged with complex-geometry
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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.
41
votes
Accepted
de Rham vs Dolbeault Cohomology
Let $\Omega^{p,q}(M)$ be the $C^{\infty}$ $(p,q)$-forms. One always has a double complex with $\Omega^{p,q}(M)$ in position $(p,q)$. The cohomology in the $q$ direction is Dolbeault cohomology, the co …
- 156k
22
votes
Accepted
Are all holomorphic vector bundles on a contractible complex manifold trivial?
No, even for line bundles. We have the short exact sequence of sheaves
$$0 \to \underline{\mathbb{Z}} \overset{2 \pi i}{\longrightarrow} \mathcal{O} \overset{\exp}{\longrightarrow} \mathcal{O}^{\ast} …
- 156k
19
votes
Accepted
Intuitive Aproach to Dolbeault Cohomology
Here is some nonsense that I find useful: On a complex manifold,
$$\frac{\mbox{locally constant functions}}{\mbox{smooth functions}} \approx \frac{\mbox{locally constant functions}}{\mbox{holomorphic …
- 156k
18
votes
Accepted
$\partial \bar{\partial}$ lemma for contractible domains
Okay, here is a counter-example. Let $X$ be the following open subset of $\mathbb{C}^2$:
$$X:= \{ (z_1, z_2) : |z_1| < 2,\ |z_2| < 1 \} \cup \{(z_1, z_2): |z_1| < 1,\ |z_2| < 2 \}$$
This is the stand …
- 156k
17
votes
Examples of smooth manifolds admitting inbetween one and a continuum of complex structures
There are countably many complex structures on $S^2 \times S^2$ up to isomorphism. Specifically, the even Hirzebruch surfaces $F_{2k}$ are the only options. This is the main result of
Qin, Zhenbo, C …
- 156k
15
votes
Accepted
A topological consequence of Riemann-Roch in the almost complex case
I believe that the displayed equation is valid for almost complex manifolds. This is closely related to a computation I talked about here.
Let $r_1$, $r_2$, ..., $r_n$ be the chern roots of the tange …
- 156k
15
votes
Accepted
Relation between the cohomology group of a curve and the cohomology group of its jacobian
$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-f …
- 156k
14
votes
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Algebraic de Rham cohomology vs. analytic de Rham cohomology
I don't think you can get this directly from GAGA. The reference that I know for this result is Grothendieck, On the de Rham cohomology of algebraic varieties. It is short, beautiful, and in English.
- 156k
13
votes
Accepted
Understanding the Hodge filtration
The naive Hodge filtration of a smooth affine variety is, indeed, the whole thing. We always have the short exact sequence of complexes:
$$0 \to \Omega^{\bullet, \geq p} \to \Omega^{\bullet} \to \Omeg …
- 156k
13
votes
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Hard Lefschetz Theorem for the Flag Manifolds
I'll spell out Hard Lefschetz as an explicit combinatorial statement about the Grassmannian $G(d,n)$. I can definitely give you a version of this for the full flag manifold if you want it and probably …
- 156k
12
votes
Accepted
Nonalgebraic complex manifolds
I wrote a blog post about some of the standard examples of nonalgebraic compact complex manifolds.
- 156k
10
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Why are local systems on a complex analytic space equivalent to vector bundles with flat con...
You might also want to read Carlos Simpson's paper "Moduli of representations of the fundamental group of a smooth projective variety", parts I and II. He explains in great detail how to make the set …
- 156k
10
votes
The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve
$\def\CC{\mathbb{C}}$I'll come back later and leave an answer to (1), which is classical and straightforward. I have nothing to say about (3). I thought the most interesting question was (2), but I am …
- 156k
10
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Converses to Cartan's Theorem B
For (4), there are finite CW complexes $X$ which are not contractible, but such that $H^q(X, \mathcal{F})$ vanishes for all (finite rank) locally constant sheaves $\mathcal{F}$. Specifically, take $X$ …
- 156k
10
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The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve
I'm getting back to the question of describing holomorphic $1$-forms on a plane curve.
Affine curves: Let $X$ be a smooth curve in $\mathbb{A}^2$, given by the equation $F(x,y)$. Then $F_x dx + F_y …