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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.
10
votes
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 $X$ …
- 40.3k
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:
$$0 \to \Omega^{\bullet, \geq p} \to \Omega^{\bullet} \to \Omeg …
- 156k
4
votes
Is this $\mathbb C$-fibration over compact Riemann surface trivial?
Maybe I misunderstood the question, but it seems to me that $M$ could be the total space of some other globally generated line bundle over $S$: For example, the total space of $\mathcal{O}(n)$ over $\ …
- 156k
6
votes
Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
The answer is no. This is a corrected version of Nicolast's comment.
Let $E$ be an elliptic curve, let $f: E \to E$ be an endomorphism and let $H_1(f) : H_1(E) \to H_1(E)$ be the induced map on $H_1$. …
- 156k
15
votes
Accepted
Relation between the cohomology group of a curve and the cohomology group of its jacobian
$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-f …
- 156k
6
votes
Accepted
When Atiyah class and Chern class coincide?
$\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$To spell out my comment a little more, let $Z^1$ be the sheaf of $\partial$-closed holomorphic $(1,0)$-forms. Since "holomorphic" means $\ov …
- 156k
8
votes
Accepted
Are smooth Schubert varieties Kähler?
Yes, smooth closed subvarieties of projective varieties are projective, and hence Kahler. Smooth Schubert varieties are very rare though, see the sources below for a description of them:
Ryan, Kevin M …
- 156k
7
votes
$H^{p,q}(X)$ versus $H^{q}(X, \bigwedge^p TX)$
One reason that $H^p(X, \bigwedge^q TX)$ will not be as well behaved as $H^p(X, \bigwedge T^{\ast} X)$ is that it is not deformation invariant, and thus not topological. In other words, if we have a c …
- 156k
9
votes
Accepted
Can all $L^2$ holomorphic functions on a domain vanish at a particular point?
$\def\CC{\mathbb{C}}$Here is a less trivial example that I think works. Let $U \subset \CC^2$ be
$$\{ (x,y) : |x| \leq \min(1, 1/|y|) \}$$
There are lots of $L^2$ holomorphic functions because the ch …
- 4,713
4
votes
Rational functions on reduced complex varieties that extend to global holomorphic functions
The answer is yes. Ariyan Javanpeykar has contributed the hard part; here are the easy parts.
Let $\tilde{A}$ be the integral closure of $A$ in $\mathrm{Frac}(A)$ and let $\tilde{X} = \mathrm{Spec}(\ …
- 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
10
votes
Accepted
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 …
- 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
4
votes
Accepted
Cotangent bundle of coadjoint orbit is stein manifold?
Not with the obvious complex structure. Notice that $O_{\lambda}$ is a closed subvariety of $T^{\ast}(O_{\lambda})$ (namely the zero section). Closed subvarieties of Stein varieties are Stein. However …
- 156k
6
votes
Are holomorphic vector bundles over Kähler manifolds Kähler
Proposition 3.18 of Voisin's Hodge Theory and Complex Algebraic Geometry I says that, if $X$ is compact Kahler and $E$ is a holomorphic vector bundle over $X$, then $\mathbb{P}(E)$ is Kahler. Since $E …
- 156k