**26**

votes

**0**answers

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi,
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...

**25**

votes

**0**answers

1k views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**18**

votes

**0**answers

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 ...

**17**

votes

**0**answers

582 views

### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

**16**

votes

**0**answers

412 views

### function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...

**12**

votes

**0**answers

310 views

### Complex manifold which is algebraic away from codimension \ge 2

If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space?
...

**11**

votes

**0**answers

295 views

### What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...

**11**

votes

**0**answers

726 views

### Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb ...

**10**

votes

**0**answers

358 views

### About the Bloch conjecture on entire curves

The Bloch conjecture states the following:
Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim ...

**8**

votes

**0**answers

222 views

### Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if
$$ du \circ j = J \circ ...

**8**

votes

**0**answers

217 views

### Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i
\geq 0$ with $\sum ik_i = n$, bounded in terms ...

**8**

votes

**0**answers

209 views

### Intersection of plus/minus cells in Bialynicki-Birula decomposition

Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:
$X = \bigsqcup_{w\in W} C_w$ ...

**8**

votes

**0**answers

742 views

### Weight filtration over the integers

This is a follow up question to Weight filtration for smooth analytic manifolds
As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...

**6**

votes

**0**answers

87 views

### What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...

**6**

votes

**0**answers

171 views

### Different complexifications of a real analytic Riemannian manifold

Hi,
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...

**6**

votes

**0**answers

254 views

### Universal property for complex blowup in smooth category

If $M$ is a smooth complex manifold and $N$ is a smooth complex surface, we may ask when a holomorphic map $f:M\rightarrow N$ lifts to a map $f:M\rightarrow [N:p]$, where $[N:p]$ denotes the blowup of ...

**6**

votes

**0**answers

502 views

### How does complex conjugation act on the Hodge decomposition?

Let $A$ be a principally polarized
abelian variety over $\mathbf{Q}$. Let $G$
be the Mumford--Tate group of $A$.
The action of complex
conjugation on $A(\mathbf{C})$ induces an involution on
the de ...

**6**

votes

**0**answers

356 views

### A non-elliptic PDE

I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X :
$\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial ...

**6**

votes

**0**answers

213 views

### Does an isomorphism between infinitesimal neighbourhoods induce a jet of a diffeomorphism?

Let $D$ be a divisor in a (compact) Kahler manifold $X$. Let $N$ be the total space of the normal bundle of $D$ in $X$ and identify $D$ with the zero section of $N$. Let $m\subset O_{X}$ and $p\subset ...

**5**

votes

**0**answers

65 views

### Density of rational functions in open Stein

I repost here, after I tried here.
Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...

**5**

votes

**0**answers

158 views

### proper mapping between Stein manifolds

My question is the following:
Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set
...

**5**

votes

**0**answers

195 views

### Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...

**5**

votes

**0**answers

166 views

### How do fibers of the functor Algebraic Varieties $\to$ Complex Analytic Spaces look like?

There's already a question (which got several interesting answers) asking about examples of the phenomenon of non (essential) injectivity of the functor $U:Alg\to AnEsp$, mapping each algebraic ...

**5**

votes

**0**answers

154 views

### Complex manifold with non-finitely generated canonical ring

P.M.H. Wilson has an example of a compact non-Kahler manifold whose canonical ring is not finitely generated; see his article and this MO question. I'm trying to understand his construction and have ...

**5**

votes

**0**answers

339 views

### Jet differentials and hyperbolicity: possible mistake in the literature?

I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...

**5**

votes

**0**answers

214 views

### About a non-degeneracy of Hodge-Riemann form..

Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let
$HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by
$HR(\alpha,\beta) = \int_M \alpha \beta ...

**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 ...

**4**

votes

**0**answers

230 views

### Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!

**4**

votes

**0**answers

136 views

### Vector bundles on Stein Manifolds

This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold X to Gr(k,n) (the Grassmannian of k planes in C^n), is there a ...

**4**

votes

**0**answers

138 views

### Closed subschemes and the analytification functor

Let $X$ be a scheme of finite type over $Spec(\mathbb C)$ and let $i:Y \hookrightarrow X$ be a closed subscheme, defined by a sheaf of ideals $J \subset \mathcal O_X$.
Then there is an induced map ...

**4**

votes

**0**answers

188 views

### Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...

**4**

votes

**0**answers

354 views

### A generalization of Liouville formula for the determinant of a system of ODE?

Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE):
$\frac{d}{dt}y= A(t) y$, where $A(t)$ is a ...

**4**

votes

**0**answers

233 views

### Is the SUSY Algebra isomorphic for all Kähler Manifolds?

For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the ...

**4**

votes

**0**answers

99 views

### Moderate growth functions on complex algebraic varieties.

Does anybody know a reference that discusses moderate growth (continuous or measurable) functions on complex algebraic varieties?
I'm interested in such a discussion in the generality of varieties ...

**4**

votes

**0**answers

260 views

### Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...

**4**

votes

**0**answers

335 views

### Atiyah--Singer for the Complex Projective Line

I'm trying to understand Atiyah--Singer by looking at the usual starting point of $CP^1$ and the Dirac--Dolbeault operator. If I've reduced everything down correctly, then in this case the theorem ...

**4**

votes

**0**answers

487 views

### Terminology for Holomorphic and Anti-Holomorphic Forms

For any complex manifold $M$ we have the usual decomposition of the cotangent bundle into the direct sum $\Omega^{(1,0)} \oplus \Omega^{(0,1)}$. I know these two sub-modules as the holomorphic and ...

**3**

votes

**0**answers

141 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 ...

**3**

votes

**0**answers

89 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. ...

**3**

votes

**0**answers

108 views

### Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...

**3**

votes

**0**answers

124 views

### Is there any advantage to knowing that Gauss-Manin is Hermitian flat?

Let $S$ be a complex manifold and let $p : E \to S$ be a holomorphic vector bundle. Is there any advantage to knowing that $E$ carries a flat Hermitian metric $h$, ie a smooth Hermitian metric with ...

**3**

votes

**0**answers

237 views

### holomorphic embeddings of the sphere into the quintic in degree 2

Is there an explicit way of
classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below)
the various families of
equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...

**3**

votes

**0**answers

144 views

### Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite global resolution by locally free modules.
By GAGA, I believe this ...

**3**

votes

**0**answers

154 views

### Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$.
...

**3**

votes

**0**answers

71 views

### Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...

**3**

votes

**0**answers

201 views

### Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...

**3**

votes

**0**answers

118 views

### Calabi-Yau structure on cotangent bundle?

Let $M$ be a smooth (compact) manifold, my question is when the cotangent bundle $T^*M$ has a Calabi-Yau structure.
Certain constructions are known, for instance, if $M=\Sigma\times S^1$ or $M$ is a ...

**3**

votes

**0**answers

143 views

### Serre duality for compactly supported sheaves

Given a smooth quasi-projective variety $X$ over $\mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P',d')$ with compactly supported cohomology. It is well-known that such complexes ...

**3**

votes

**0**answers

106 views

### On the Haar measure of Grassmanians

How can one write down the Haar measure of complex Grassmanians in terms of Plucker coordinates? Is there any way to define a Kahlarian measure like $d\mu\propto \det(g)dp_{ij}dp^{*}_{ij}$ where $g$ ...

**3**

votes

**0**answers

221 views

### A question on fibered Calabi-Yau threefolds

Let $\phi:X\rightarrow \mathbb{P}^1$ be a fibered Calabi-Yau threefold with a general fiber $F$. The following are known
$\phi=\Phi_{mF}$ for some $m\in \mathbb{N}$, where $\Phi_D$ stands for the ...