**0**

votes

**1**answer

157 views

### what is the first cohomology group of structure sheaf of grassmannian [closed]

I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.

**8**

votes

**1**answer

304 views

### Hodge numbers of diffeomorphic complete intersections

Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers?
Edit: as written by Daniel Loughran in the comments below, complete ...

**4**

votes

**1**answer

407 views

### What is the automorphism of a grassmannian?

I want to know what is the holomorphic automorphism of a grassmannian. Can someone tell me this?

**2**

votes

**1**answer

82 views

### Projectively flat Hermitian curvature proportional to Kähler form?

Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to ...

**5**

votes

**1**answer

251 views

### A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let ...

**0**

votes

**0**answers

88 views

### complementary bundle for a divisor

For a divisor in a complex manifold, what is known about a complementary bundle to the divisor in the manifold (either for the tangent or the cotangent bundle). Is there a description in terms of ...

**2**

votes

**1**answer

62 views

### Show properness of Ahlfors map

If have got a $k$-fold connected surface $G$, which is bounded by n distinct, non-intersecting Jordan-curves. By Ahlfors it is known that there exists a unique function $\phi$ which maps G to the unit ...

**5**

votes

**1**answer

478 views

### Kodaira-Spencer theory of deformation done right

I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...

**4**

votes

**1**answer

137 views

### Simple maps: Flat versus locally trivial

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...

**0**

votes

**1**answer

81 views

### Condition for infinite dimensional complex manifold to be Kähler by pullback form

For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced ...

**4**

votes

**1**answer

167 views

### Non Kähler blow-up of a Kähler manifold

Is it possible to find a complete, non compact Kahler manifold $(X,\omega)$ with a closed, connected, non compact complex submanifold $Y\subset X$ of codimension at least 2 such that the blow-up of ...

**2**

votes

**2**answers

268 views

### Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps?

Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J_N$ and $J_M$).
Does it imply ...

**10**

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, ...

**7**

votes

**1**answer

218 views

### Extending holomorphic functions

Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on ...

**1**

vote

**0**answers

137 views

### Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...

**6**

votes

**1**answer

146 views

### Is the total space of a family of normal varieties a normal variety?

Let $f:X \rightarrow C$ be a flat morphism from a complex variety $X$ to a smooth curve $C$. If any fiber $X_{t}=f^{-1}(t)$ is a reduced normal projective variety, is the total space $X$ a normal ...

**0**

votes

**0**answers

118 views

### classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...

**4**

votes

**1**answer

150 views

### Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions)

Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I ...

**3**

votes

**2**answers

219 views

### Families of Fano varieties over non-hyperbolic curves

Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...

**2**

votes

**0**answers

90 views

### Chern-Weil theory for degenerated metric

If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of ...

**7**

votes

**1**answer

428 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 ...

**1**

vote

**1**answer

111 views

### characterization of structure group

Somebody tell me that:
For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power.
Assume that the ...

**1**

vote

**0**answers

339 views

### What implications would a solution of the *Standard Conjectures* have on the *Hodge Conjecture*?

I'm new to the field, so I just would like to know what implications would have a solution of the Standard Conjectures on the Hodge Conjecture. I read somewhere they are related in some way, but I ...

**1**

vote

**1**answer

175 views

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

Yesterday I asked the following question to which abx has given a positive answer.
examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes
But I suddenly realized ...

**0**

votes

**1**answer

291 views

### A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding
$$\phi: (M,\omega)\to (\mathbb CP^N, ...

**2**

votes

**1**answer

246 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 ...

**1**

vote

**0**answers

48 views

### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...

**6**

votes

**1**answer

165 views

### Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, thus
$$3\tau+2\chi=c_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...

**0**

votes

**1**answer

73 views

### Meromorphic extension of local defining equations of a complex submanifold

let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can ...

**4**

votes

**1**answer

69 views

### Minimal projective space containing projective variety independent of base field

In this question I ask whether ambient spaces descend to models of varieties.
Let $k\subset K$ be a non-trivial extension of algebraically closed fields, e.g., $\overline{\mathbb Q}\subset \mathbb ...

**5**

votes

**2**answers

421 views

### Are “ample” and “positive” line bundle the same concept?

A line bundle is ample if some power of it is very ample. A line bundle is positive if the chern class in $H^2(X,\mathbb{Z})$ is represented by a Kahler metric in $H^{1,1}(X,\mathbb{Z})$.(Regarded as ...

**2**

votes

**1**answer

130 views

### Bound for the Picard number of a 3-fold

Are there some upper bounds for the Picard number of a non-singular threefold? We know that in the surface case, we have $h^{1,1}=10\chi-c_1^2+2q$. Hence the picard number $\leq 10\chi-c_1^2+2q$. Is ...

**0**

votes

**0**answers

79 views

### Commuting Derivative and Convolution type integral

Suppose $\Gamma$ is a smooth curve, $f$ and its derivative belong to some $L^p(\mathbb{C})$(i.e $f\in W^{1,p}$) and kernel $K(|z-y|)\in\mathbb{C}\times\mathbb{C}$ has only singularity on ...

**0**

votes

**0**answers

38 views

### Expansion of the squared distance function from a submanifold

let $(M,g)$ be a compact Kahler manifold of dimension $m$ with the Kahler metric $g$ real-analytic. Let $N\subset M$ be a complex submanifold of dimension $0<k<m$. Let $$d_{g}(M,\cdot): ...

**3**

votes

**0**answers

90 views

### Ampleness of Hodge bundles over complex curves

Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. ...

**2**

votes

**0**answers

46 views

### Characterisation of convergence in Deligne-Mumford compactifiaction

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...

**1**

vote

**1**answer

61 views

### Deformations of a pair of compact, complex manifolds

This is somewhat related to the question found at What is the DGLA controlling the deformation theory of a complex submanifold?, though not exactly the same, so I hope it's not duplicating too much. ...

**4**

votes

**2**answers

275 views

### Must a canonical line bundle be associated to a cartier divisor?

Suppose $X$ is a complex manifold, we have the map $H^0(X,K_X^*/O_X^*)\to H^1(X,O_X^*)$, is the canonical line bundle $\wedge^n{\Omega}$ always in the image of the map?

**0**

votes

**0**answers

279 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 ...

**6**

votes

**1**answer

202 views

### Is the number of minimal models finite

Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...

**4**

votes

**0**answers

124 views

### Degeneration of varieties ---Mumford reduction---minimal model, what's the relationship?

Every variety here is over $\mathbb{C}$.
Let $f: X \rightarrow C $ be a flat proper surjective morphism from a quasi-projective variety $X$ to a quasi-projective smooth curve. Let $p\in C$. We ...

**2**

votes

**0**answers

162 views

### When is the analytification of a variety homeomorphic to $\mathbb C$

Let $X$ be the analytification of a variety $V$ such that $X$ is homeomorphic to $\mathbb R^2$.
What can we say about $X$? Can $V$ be singular?

**1**

vote

**1**answer

81 views

### How far do conjugated Mobius transforms move points?

I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - ...

**1**

vote

**3**answers

167 views

### Lattice polarized K3 surfaces

I've recently encountered the definition of a lattice polarized K3 surface. What is the idea behind the definition? Surely, there's something deeper to it than merely being a natural generalization of ...

**3**

votes

**1**answer

147 views

### Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...

**13**

votes

**1**answer

623 views

### Are all holomorphic vector bundles on a contractible complex manifold trivial?

It is true that over a contractible manifold all differentiable vector bundles are trivial. However the method of proof does not apply in the holomorphic category.
It is also true that a contractible ...

**2**

votes

**0**answers

173 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

**3**

votes

**1**answer

116 views

### Harmonic function osculating a given subharmonic function

Let $\Omega\subseteq\mathbb C$ be an open set and let $\phi:\Omega\to\mathbb R_{\geq 0}$ be an exhausting (i.e. proper) smooth subharmonic function. Fix $p\in\Omega$. Does there exist a harmonic ...

**4**

votes

**1**answer

415 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

**1**answer

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 $f(\gamma)$ ...