7
votes
1answer
429 views

Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know Let $(M,\omega)$ be a compact ...
4
votes
2answers
451 views

A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...
2
votes
1answer
174 views

The behaviour of holomorphic mapping of curves

Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ ...
0
votes
0answers
71 views

moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
0
votes
0answers
110 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
0
votes
1answer
211 views

A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$. Question: Is the connectivity of ...
-2
votes
2answers
229 views

topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology. I am not sure if the following two results are true, and where can I find the references: (1) ...
2
votes
1answer
115 views

Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$. If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...
2
votes
1answer
278 views

On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
5
votes
1answer
321 views

Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
0
votes
1answer
269 views

question about the developing map

I'm having some trouble finding literature on the developing map. All the sources I could find on it seem to refer to thurston's definition in either: http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf or ...
6
votes
3answers
304 views

Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...
2
votes
2answers
195 views

Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site. Let $S$ be some orientable surface obtained by removing ...
4
votes
1answer
604 views

an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact. we know that ...
1
vote
0answers
102 views

Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group and $\Gamma$ a discrete subgroup of $G$ such that the subgroups $\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...
3
votes
0answers
199 views

Good covers on complex algebraic varieties with normal crossings singularities

Let $X$ be a topological space. A good cover on $X$ is an open cover such that all finite non-empty intersections are contractible. It is a theorem of Hironaka that (complex) algebraic sets admit ...
3
votes
1answer
621 views

Connections on line bundles over the torus

If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles ...
3
votes
2answers
683 views

book on calabi yau manifolds

hi, does anybody know a good book on calabi yau manifolds (i am a beginner) ? thanks in advance lois
12
votes
1answer
567 views

Are surface bundles over a surface with non-zero signature necessarily complex (or algebraic)?

By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature ...
12
votes
3answers
1k views

A four-dimensional counterexample?

Does anyone know an example of a smooth hyperbolic surface bundle over a hyperbolic surface (surface = compact two-manifold) which does not have a complex structure? Is there any decision procedure to ...
8
votes
2answers
483 views

Small neighborhoods of singularities on varieties

In Singular points of complex hypersurfaces, John Milnor proves the following theorem: Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...
18
votes
0answers
668 views

Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$? The following sub question is ...
13
votes
4answers
1k views

Representations of surface groups via holomorphic connections

EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it! Background Let $E \to X$ be a ...