**9**

votes

**1**answer

381 views

### Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...

**3**

votes

**1**answer

221 views

### Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...

**2**

votes

**2**answers

256 views

### Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...

**1**

vote

**1**answer

295 views

### Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity

Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have ...

**81**

votes

**5**answers

4k views

### Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer,
moving initially $\pm$ in the horizontal
or vertical directions. When it hits a Gaussian prime, it turns left ...

**29**

votes

**5**answers

3k views

### Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...

**22**

votes

**5**answers

3k views

### References for “modern” proof of Newlander-Nirenberg Theorem

Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...

**15**

votes

**5**answers

2k views

### Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...

**22**

votes

**5**answers

2k views

### Two definitions of Calabi-Yau manifolds

Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle ...

**26**

votes

**2**answers

2k views

### A geometric characterization for arithmetic genus

Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...

**18**

votes

**4**answers

3k views

### Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...

**19**

votes

**1**answer

1k views

### Diffeomorphic Kähler manifolds with different Hodge numbers

This question made me wonder about the following:
Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?
It seems that this would require that those manifolds are not ...

**15**

votes

**4**answers

735 views

### Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'.
Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...

**22**

votes

**3**answers

831 views

### Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...

**12**

votes

**3**answers

678 views

### What is the DGLA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. ...

**10**

votes

**2**answers

515 views

### A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...

**5**

votes

**3**answers

873 views

### When is a holomorphic submersion with isomorphic fibers locally trivial?

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a ...

**6**

votes

**3**answers

1k views

### Lefschetz Hyper-plane theorem for singular projective varieties?

Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says:
For smooth hyperplane section $Y= X\cap H$, the restriction map
$H^i(X) \rightarrow H^i(Y)$ is an ...

**6**

votes

**3**answers

1k views

### Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let X be a complex analytic space. It is a 'well known fact' that the categories of local systems on X (i.e. locally constant sheaves with stalk C^n), and of (holomorphic) vector bundles on X with ...

**15**

votes

**1**answer

1k views

### Almost Complex Structure approach to Deformation of Compact Complex Manifolds

I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...

**4**

votes

**1**answer

732 views

### Atiyah class for non-locally free sheaf

Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$.
The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the ...

**12**

votes

**5**answers

663 views

### Comparing fundamental groups of a complex orbifolds and their resolutions.

Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
...

**11**

votes

**2**answers

1k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see : Most interesting mathematics mistake?
Added : According to their method, ...

**11**

votes

**2**answers

611 views

### Vector bundles on vector bundles

Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X ...

**8**

votes

**1**answer

957 views

### Negative holomorphic sectional curvature

Let X be a complex hermitian manifold with hermitian form $\omega$. How can you prove that if $\omega$ has negative holomorphic sectional curvature, then its scalar curvature is negative, too?

**6**

votes

**1**answer

482 views

### Blowing-up an ordinary double point, then contracting the exceptional locus to a curve

Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...

**2**

votes

**2**answers

269 views

### Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be ...

**7**

votes

**2**answers

325 views

### Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...

**6**

votes

**3**answers

1k views

### Question about Hodge number

Hi. I am studying Hodge theory on Kahler manifolds.
I have several questions.
Is Hodge number a topological invariant? (I mean, is it independent of the choice of
Kahler structure?)
If the ...

**6**

votes

**3**answers

645 views

### Newlander-Nirenberg for surfaces

Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...

**2**

votes

**1**answer

460 views

### Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...

**5**

votes

**1**answer

324 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 ...

**5**

votes

**2**answers

501 views

### On the fundamental group of hypersurfaces

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})=
...

**4**

votes

**1**answer

182 views

### Extending Functions on Closed Submanifolds of C^n

Functions on an algebraic subvariety X of A^n are the same as functions on A^n restricted to X. So the statement that functions on X extend to all of A^n follows by the definition. My question is: ...

**3**

votes

**2**answers

559 views

### Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifold of projective space?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times ...

**2**

votes

**1**answer

250 views

### examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$，where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and ...

**0**

votes

**0**answers

283 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**0**

votes

**1**answer

231 views

### Can one embedd the projectivezed tangent space of CP^2 in a projective space?

Given a complex vector bundle $V\rightarrow M$, we can form a
fibre bundle $\mathbb{P} V\rightarrow M$, where the fiber over
each point is the corresponding projective space. In particular
consider ...

**10**

votes

**3**answers

414 views

### orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily ...

**8**

votes

**1**answer

260 views

### counterexample to the Chern number inequality on Fano manifold

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality
$$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$
My question is whether there ...

**3**

votes

**1**answer

220 views

### Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know:
1) if its second Betti number is non zero,
2) same question but now $V$ is a ...

**3**

votes

**1**answer

184 views

### General position argument for reasonable spaces

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p ...

**2**

votes

**1**answer

202 views

### Local expression involved in the definition of positivity of vector bundles

This is question follows on from this one.
In the linked question, the hermitian form $\theta_E$ on $T^{1,0}X\otimes E$ is defined globally as ...

**1**

vote

**1**answer

254 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 ...

**0**

votes

**1**answer

96 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 ...

**0**

votes

**2**answers

301 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?