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Results tagged with complex-geometry
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user 2349
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.
4
votes
Accepted
Holomorphic map from a neighborhood in $\mathbb C$ to S^3
Suppose such a map exists and is non-constant. Let $f,g$ be its components. By composing the map with a holomorphic function on the right and with an element of $U(2)$ on the left, we can assume that …
- 24.6k
13
votes
Does there exist a complex Lie group G such that ...
Let me show that such a group, if it exists, can't be algebraic. Any complex algebraic group $G$ is an extension of an abelian variety $A$ by an affine group $H$, i.e. we have an exact sequence $1\to …
- 24.6k
4
votes
Strong Kodaira vanishing
Re question 1: yes, see Deligne, Cohomologie des intersections compl`etes, SGA 7 II, th\'eor`eme 1.1. The proof is by "force brutale" as Deligne himself puts it, so I'm not sure this generalizes.
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10
votes
1
answer
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Extending holomorphic functions
Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on the …
- 24.6k
7
votes
1
answer
558
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Hodge theory for quasi-Kaehler manifolds: where does it break down?
Let $U$ be the complement of a divisor with normal crossings in a smooth compact complex manifold $X$. If $X$ is algebraic, then Deligne describes in "Th\'eorie de Hodge 2" a procedure to equip the ra …
- 24.6k
4
votes
Lefschetz hyper-plane theorem for singular projective varieties?
I think what you are looking for is the Lefschetz hyperplane theorem for singular varieties from Goresky and MacPherson's " Stratified Morse theory" (part II, section 1.2). The range in which there is …
- 24.6k
2
votes
Complex analytic space with no (positive dim.) subscheme ?
Take a complex 2-torus $X$ without curves, and hence, without non-constant meromorphic functions (see e.g. Shafarevich, Basic algebraic geometry, chapter VIII, \S 1, example 2). The only locally close …
- 24.6k
2
votes
Accepted
On bounded homogeneous connected domains of C^n
Re question 3: a bounded homogeneous domain is biholomorphic to a Siegel domain, which is contractible. See e.g. Siegel domain and references therein (those references probably answer question 2 as we …
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2
votes
Accepted
Does passing through a point in general position cut down the dimension by one?
Q1: In general no: take $X$ to be the subvariety of $P^{\delta_d\vee}$, the dual of $P^{\delta_d}$, formed by all $H_p$'s. Note that $X$ is just the image of the Veronese map $\mathbb{P}(V)\to\mathbb{ …
- 24.6k
16
votes
Is there a complex structure on the 6-sphere?
If such a complex structure exists, it would weird indeed! For example, as shown by Campana, Demailly and Peternell (Compositio 112, 77-91), if such a thing exists, then $S^6$ would have no non-consta …
5
votes
Nonalgebraic complex manifolds
The simplest example of a complex analytic non-algebraic manifold (and hence, non-projective) is probably the Hopf surface. Indeed, any smooth complete complex algebraic variety, projective or not, is …
- 24.6k
7
votes
Algebraic de Rham cohomology vs. analytic de Rham cohomology
This does follow from GAGA via the spectral sequences associated to the dumb filtrations on the algebraic and analytic de Rham complexes of sheaves, see p. 96 of tome 29 of PMIHES in a paper of Grothe …
- 24.6k
19
votes
4
answers
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A topological consequence of Riemann-Roch in the almost complex case
This question originated from a conversation with Dmitry that took place here
Is there a complex structure on the 6-sphere?
The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of …
- 24.6k
10
votes
0
answers
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Weight filtration over the integers
This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a wei …
- 24.6k
4
votes
Accepted
Relation between sheaf and group cohomology
I doubt that in general one can construct a reasonable sheaf on $U$ with the required properties. To see what kind of bad things can happen, let us try to understand why this works for $X$ an elliptic …
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