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Results tagged with complex-geometry
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user 1345
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.
3
votes
Continuous point map for spherical domains
Here’s an attempt to define such a map which is not an answer but couldn’t fit in a comment.
Consider a Riemann mapping $\varphi:\mathbb{H}^2 \to \overset{\circ}{D}$ (which has a continuous extension …
- 68.9k
12
votes
Accepted
Symplectic structure on the square of a 3-manifold
Let $M$ be a 3-manifold fibering over $S^1$, so there exists a fibration $\Sigma \to M \to S^1$.
Then $M\times M$ will admit a symplectic structure.
There is a symplectic structure on $M\times S^1$, a …
- 68.9k
31
votes
Accepted
Complex projective manifolds are homeomorphic if homotopy equivalent
For curves this follows from the classification of (2-dimensional topological) surfaces, and for simply-connected surfaces this follows from Freedman's theorem.
My former colleagues Anatoly Libgober a …
- 68.9k
9
votes
Complexifications of hyperbolic manifolds
There exist hyperbolic 3-manifolds which cannot embed totally geodesically in complex hyperbolic manifolds, answering this
question in the negative.
Recently it was shown that complex hyperbolic manif …
6
votes
Accepted
On limits of manifolds
In general, this will be false. Examples are found among solenoidal manifolds, defined by Sullivan. For example, 1-dimensional solenoids.
Many of these are obtained by taking the inverse limit of fi …
- 68.9k
3
votes
Lattices of PU(n,1) with large abelianization
The other answers give fine examples, but I found a reference to the 1981 thesis of Livné which constructs complex hyperbolic lattice which admits a surjective holomorphic map to a Riemann surface. Th …
- 68.9k
2
votes
When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?
If $X$ or $Y$ is $\mathbb{D}^n$ (the unit complex ball) or $\mathcal{T}^n$ (Teichmuller space), then $Aut(X\times Y)$ will be finite index in $Aut(X)\times Aut(Y)$. This follows by considering the Kob …
- 68.9k
6
votes
Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?
There's a remarkable theorem of Margulis that pertains to your question. Let $G$ be a semisimple Lie group (in your case, $PSL_2(\mathbb{R})$), and let $\Gamma$ be an irreducible lattice in $G$. The c …
- 68.9k
5
votes
Surface of a Ideal Tetrahedron in Hyperbolic Space H3
The surface of the tetrahedron is a 4-punctured sphere, made of 4 ideal triangles. In fact, this is a complete hyperbolic metric, as may be seen by sending $z_1$ to $\infty$ (in fact, we may assume $z …
- 68.9k
3
votes
Injective maps on cohomology and Kahler manifolds
Suppose one has compact symplectic manifolds $(X,\omega), (Y,\sigma)$, and a map $f:X\to Y$ such that $f^\ast\sigma=\omega$ (if $f$ is also a diffeomorphism, then this map is a symplectomorphism, but …
- 68.9k
12
votes
Accepted
A four-dimensional counterexample?
This paper by Hillmann addresses this question. He proves that a surface bundle over a surface which is a complex surface has a holomorphic fibration over the base, for some choice of complex structur …
- 68.9k
6
votes
Kähler structure on cotangent bundle?
In a paper by Goldman, Kapovich, and Leeb, it is pointed out that a fuchsian (surface) group embedded into the isometries of complex hyperbolic space has quotient the tangent bundle to the surface. Si …
- 68.9k
6
votes
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
I think the short answer to your question is no, since the notion of a holomorphic map is purely local, whereas the existence of a hyperbolic metric (uniformization) depends on the global structure of …
- 68.9k
5
votes
Rational Hilbert modular surfaces
I don't know the answer to this question, but here's a possible strategy to prove finiteness, following the approach of Zograf, as extended by Long and Reid for congruence arithmetic fuchsian groups o …
- 68.9k
36
votes
What do Weierstrass points look like?
I think another interpretation is in terms of Euclidean geometry instead of hyperbolic geometry. If $\omega$ is a holomorphic 1-form on a Riemann surface $\Sigma$, then $|\omega|$ defines a Euclidean …
- 68.9k