**0**

votes

**0**answers

77 views

### Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds.
My question is:
Is the group of symplectic automorphisms
of $S^2$ with respect to this ...

**8**

votes

**2**answers

421 views

### Finite etale atlas for Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.
Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?
What if $X$ is an algebraic space ...

**1**

vote

**1**answer

129 views

### Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...

**5**

votes

**0**answers

103 views

### Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind.
Let $X$ be a smooth quasi-projective geometrically ...

**5**

votes

**1**answer

96 views

### Special Kähler normal coordinates around a point

Let $(M,\omega)$ be a compact Kähler manifold and suppose there are holomorpic vector fields vanishing at a point $p$. As a consequence we have a group $G_{p}$ of biholomorpisms fixing $p$. Let ...

**2**

votes

**2**answers

141 views

### Holomorphic Line Bundles over a Homogeneous Space

Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...

**0**

votes

**0**answers

88 views

### G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...

**0**

votes

**1**answer

81 views

### Location of the zeros set of holomorphic function [closed]

Recently I proved the following result.
"If a holomorphic function $f$ maps the unit disc $\Delta$ into the unit disk $\Delta $ with $0<|f(0)|$ then $f$ doesn't vanish in the disk $D(0,|f(0)|)$. "
...

**6**

votes

**1**answer

257 views

### Can we normalize a complex analytic space in a covering of an open subset?

Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that ...

**3**

votes

**1**answer

182 views

### Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...

**0**

votes

**0**answers

186 views

### Holomorphic vector bundles over $\mathbb{CP}^1$ and elliptic curves

Can anyone suggest a good exposition of the classification of holomorphic vector bundles over $\mathbb{CP}^1$? Also does there exist any analytic or more elementary proof of Atiyah's classification of ...

**4**

votes

**0**answers

77 views

### Looking for some abelian surface fibration

Do you know of any explicit smooth complex projective threefold $X$ with an Abelian surface fibration over $\mathbb{P}^1$ such that $K_X = [-F]$ where $F$ is the fiber class divisor.
I am not ...

**3**

votes

**2**answers

427 views

### Mirror Symmetry for Quaternionic-Kähler Manifolds

I take the following quote from Huybrecht's notes on hyperkähler manifolds and mirror symmetry:
Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of ...

**0**

votes

**1**answer

104 views

### Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$.
Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k) $ be the complexified Lie group of $Z(k)$ ...

**3**

votes

**0**answers

136 views

### different proofs of the fact that compact riemann surface has a non-trivial meromorphic function

I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...

**8**

votes

**1**answer

215 views

### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

**1**

vote

**1**answer

215 views

### Blow-ups and cohomology

I'm trying to understand how to compute the Chow ring of a blow-up.
Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. ...

**3**

votes

**0**answers

104 views

### Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...

**1**

vote

**1**answer

97 views

### Nonpositive curvature of Stein manifolds

It is a theorem of Greene and Wu that a complete, simply-connected Kaehler manifold of everywhere nonpositive sectional curvature is a Stein manifold. I am curious about what kinds of additional ...

**0**

votes

**1**answer

97 views

### curve through a point avoiding an hypersurface

Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve ...

**1**

vote

**1**answer

160 views

### The locus of rational/elliptic curves on a special surface in $\mathbb{P}^3$

Let $P$ and $Q$ be two general polynomials of the same degree $d>5$. Consider the surface $S: z^2=P(x)Q(y)$ in $\mathbb{P}^3$ (after homogenization by the variable $w$). One can show that these ...

**4**

votes

**2**answers

208 views

### A question on certain elliptic PDE

Consider the elliptic PDE "CR"
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are similar to the Cauchi ...

**1**

vote

**0**answers

111 views

### Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...

**1**

vote

**0**answers

91 views

### DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...

**0**

votes

**1**answer

86 views

### Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.
(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)
Assume we ...

**2**

votes

**2**answers

229 views

### Holomorphic vector fields on blow-ups of CP^2

On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone ...

**0**

votes

**1**answer

123 views

### Hilbert scheme of a closed subscheme

Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over ...

**1**

vote

**1**answer

275 views

### Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...

**1**

vote

**1**answer

125 views

### Hilbert scheme of an infinitesimal neighborhood of a subvariety

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...

**0**

votes

**0**answers

62 views

### Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:
$X$ has a finite number of singularities on $S^{2}$
The equator is invariant under flow of $X$
3.$g_{*}X=X\; ...

**2**

votes

**1**answer

179 views

### Does the “Ohsawa-Takegoshi theorem without bounds” have a name?

There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...

**3**

votes

**0**answers

158 views

### Uniruled degenerations of abelian varieties

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...

**2**

votes

**2**answers

331 views

### All Kähler metrics on a complex manifold?

Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$?
For the case of surfaces ($dim_C=1$), ...

**0**

votes

**0**answers

63 views

### Determinant of an action and characters

In the paper of Ramanathan "Stable Principal Bundles on a Compact Riemann Surface", I read: ...where $\mu$ is the determinant of the (adjoint) action of $P$ on ...

**1**

vote

**0**answers

110 views

### The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...

**4**

votes

**1**answer

163 views

### Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...

**2**

votes

**1**answer

95 views

### Deformation of Canonical singularities

Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X_{t_{0}}=p^{-1}(t_{0})$, for a $t_{0}\in T$, has only canonical singularities of index $1$. ...

**4**

votes

**1**answer

140 views

### Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...

**1**

vote

**0**answers

42 views

### extending to bimeromorphic maps

A meromorphic map of complex spaces (in the sense of Remmert) f:X→Y is a multivalued map such that its graph Γ is an analytic subset of X×Y and off some analytic subset Z⊂Γ, the projection on the ...

**2**

votes

**0**answers

114 views

### Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...

**2**

votes

**1**answer

257 views

### Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...

**2**

votes

**2**answers

241 views

### Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...

**2**

votes

**2**answers

133 views

### Where can I find a translation of Caspar Wessel's “Om directionens analytiske betegning?”

I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled On the Analytical Representation of ...

**2**

votes

**0**answers

85 views

### Complex structure on the set of prequantization line bundles

For geometric quantization, the set of equivalence classes of prequantization line bundles of a quantizable symplectic manifold $(M, ω)$ is parametrized by $H^1(M, S^1)$ which represents the ...

**-2**

votes

**1**answer

92 views

### Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...

**2**

votes

**0**answers

108 views

### If a compact real submanifold of $\mathbb{CP}^n$ is approximable by complex algebraic curves, is it algebraic?

To make this into a separate question:
If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to ...

**3**

votes

**0**answers

220 views

### Complex structures on Riemann surfaces

This is cross posted from math.SE: http://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a map ...

**1**

vote

**0**answers

96 views

### A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof?
Statement. ...

**0**

votes

**0**answers

62 views

### Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension?
I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is
$$
...

**17**

votes

**2**answers

749 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...