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Results tagged with complex-geometry
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user 519
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.
27
votes
Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
Let $E$ be an elliptic curve. The moduli space $M_E$ of line bundles with a connection on $E$ is an $\mathbb{A}^1$ bundle over $Pic^0(E) \cong E$. In particular, $E$ can be recoved from $M_E$ as the A …
- 10.5k
11
votes
Accepted
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
I combine user37314's answer and my comments; the claim is that any smooth projective complex algebraic surface with $\pi_1$ abelian and $\pi_2$ finite has a finite cover which must be an abelian surf …
9
votes
Accepted
Are any of these complex surfaces ever projective?
Here is a simple method for constructing projective examples:
Assume there exist maps $f:C \to \mathbb{P}^1$ and $g:T \to \mathbb{P}^1$ of the same degree which are totally ramified at $c$ and $t$. Le …
7
votes
Accepted
Comparing fundamental groups of a complex orbifolds and their resolutions.
A reference is Theorem 7.8 of the article by Kollar: "Shafarevich maps and plurigenera of algebraic varieties", Invent. Math. 113. This proves the equality of fundamental groups for quotient singualri …
- 10.5k
5
votes
Accepted
diffeomorphic, holomorphic, biholomorphic
Yes. Since the map is a diffeomorphism onto its image it means that the induced map on tangent spaces (thought of as $C^{\infty}$ manifolds) is an injection. But the tangent space as a complex manifol …
- 10.5k
5
votes
Accepted
Infinitesimal deformations of fake projective planes (or ball quotients)
Yes, this is true. It follows from a more general theorem of Calabi and Vesentini on the vanishing of $H^i(X, T_X)$ (in a suitable range) for $X$ a compact quotient of an irreducible Hermitian symmetr …
4
votes
Accepted
Extending Functions on Closed Submanifolds of C^n
Yes, this is true. It follows from "Cartan's Theorem B" which says that H^1 of any coherent analytic sheaf on a closed submanifold of C^n is 0; the same result is also true for analytic subspaces. Loo …
- 10.5k
4
votes
Accepted
Is the Baily--Borel compactification functorial?
A useful reference might be the article "Satake Compactification and extension of Holomorphic Mappings", Inv.Math. 16, 237-248, 1972, by Kiernan and Kobayashi. They show that if the map $V_1 \to V_2$ …
- 10.5k
2
votes
Accepted
On morphisms of pure Hodge structures of decreasing weight
Here is a proof of the Scholie:
For a pure Hodge structure $H$ of weight $n$ we have $H_{\mathbb{C}} = \oplus_{p+q=n} H^{p,q}$ where we have $H^{p,q} := F^p \cap \bar{F}^q$. The key point is that $H^ …
- 10.5k
2
votes
Accepted
What can be said about a projective morphisms that admit decomposition theorem like smooth m...
Here is an example where $f$ is not smooth but $Rf_* \mathbb{C}$ behaves as if it were:
Let $X$ be a hyperelliptic surface and $f$ the natural morphism to $Y \cong\mathbb{P}^1$. All reduced fibres of …