**3**

votes

**1**answer

168 views

### A weak analytic version of the valuative criterion of properness

EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...

**11**

votes

**1**answer

319 views

### Thom conjecture in CP3

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...

**1**

vote

**2**answers

157 views

### there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H

"Suppose that X is a compact projective manifold
equipped with a K¨ahler metric ω. Let L be a holomorphic line bundle
In general, there exists a hypersurface H ⊂ X such that
X \ H is Stein and L is ...

**0**

votes

**1**answer

171 views

### existence of positive curved line bundles on a compact Riemann surface

Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...

**3**

votes

**1**answer

229 views

### Discriminant of a singular conic bundle

In the context of the Minimal Model Program it can arise that we need to deal with contractions of extremal rays that are conic bundles $\pi:X\to Y$ with relative Picard number 1 (with possibly ...

**0**

votes

**0**answers

67 views

### A question about bergman kernel

This is lemma 3.3 in B.Berndtsson's paper "Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains":
Let $\Omega$ be a bounded domain and $\phi_j$ a ...

**4**

votes

**2**answers

208 views

### Projective curves of constant curvature

A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...

**7**

votes

**0**answers

171 views

### Complex structures on $\Bbb R^4$

Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...

**4**

votes

**1**answer

259 views

### Torsors in the analytic topology versus torsors in the etale topology

Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over ...

**0**

votes

**0**answers

45 views

### on the existence of holomorphic coordinates under bounded curvature

Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...

**0**

votes

**0**answers

122 views

### Kodaira-Spencer theory in d=1

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$.
To determine dimension of $T\mathcal M_g$,
start with a complex structure, which in some coordinates can be written
...

**1**

vote

**1**answer

121 views

### Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...

**0**

votes

**0**answers

67 views

### A question od being algebraic stable for birational map

Recently I need to read a paper related to complex algebraic geometry and several complex variable. I think I may need some criterion of a function being algebraic stable.
Let $f$ be a birational ...

**3**

votes

**0**answers

155 views

### Properties of finite quotients of quasi-projective varieties

Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...

**3**

votes

**0**answers

141 views

### Toroidal compactifications

Does anyone know if there is an intrinsic definition of a toroidal compactification (over $\mathbb{C}$)?
Something like: Let $X$ be an algebraic variety over $\mathbb{C}$. Then $X \subset \bar{X}$ ...

**3**

votes

**2**answers

484 views

### Different definitions of spin structures

This is the definition of spin structure according to Wikipedia:
which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one ...

**0**

votes

**0**answers

136 views

### kahler manifolds with positive holomorphic sectional curvature

It is well known that a compact Kahler manifold with positive holomorphic bisectional
curvature is biholomorphic to $CP^n$. However, if we just assume positive holomorphic
sectional curvature, is ...

**5**

votes

**0**answers

63 views

### Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...

**0**

votes

**1**answer

223 views

### Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper
In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...

**1**

vote

**0**answers

235 views

### On the Hitchin fibration

I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...

**1**

vote

**0**answers

121 views

### Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...

**1**

vote

**0**answers

89 views

### Extending integrable almost-complex structure

Suppose $(X,I)$ is an almost-complex real analytic manifold where $I$ is a real-analytic almost complex structure. Suppose there exists an $I$-almost complex submanifold $M\subset X$ where this means ...

**2**

votes

**2**answers

292 views

### Is there an Oka-Grauert principle for homogeneous spaces?

Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured ...

**1**

vote

**1**answer

222 views

### Monodromy of a punctured disc

I meet the following problem which I think related to the monodromy:
Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, ...

**15**

votes

**3**answers

418 views

### Can all $L^2$ holomorphic functions on a domain vanish at a particular point?

Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space ...

**4**

votes

**1**answer

110 views

### $L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} ...

**0**

votes

**0**answers

72 views

### About some 'rigidity theorem' for the Kahler forms on projective bundles

Let $E\to X$ be a holomorphic vector bundle over a compact Kahler manifold $X$ with Kahler form $\omega_{X}$. For a given hermitian metric on $E$, let $\omega_{E}$ be the Chern form of the line bundle ...

**0**

votes

**0**answers

111 views

### When can one find holomorphic sections vanishing at a point to a certain order?

Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...

**6**

votes

**1**answer

230 views

### Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...

**6**

votes

**1**answer

224 views

### Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:
Find a pair of complex analytic families $\lbrace M_t\rbrace$ and ...

**10**

votes

**1**answer

750 views

### Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728

So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not ...

**1**

vote

**1**answer

149 views

### On holomorphic vector bundles over compact Kahler surfaces

Let $E\to X$ be a complex vector bundle over a compact Kahler surface $X$. Assume $c_{i}(E)\in H^{i,i}(X)$ for all i. Does the bundle $E$ admit a holomorphic structure?

**1**

vote

**0**answers

87 views

### Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold.
Suppose $Y$ is a conifold transition on a single nullhomologous
Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to
...

**8**

votes

**0**answers

246 views

### Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...

**4**

votes

**0**answers

114 views

### Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...

**2**

votes

**1**answer

157 views

### For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} - 1$. Do we need to use the Enriques-Kodaira classification?

In the Wikipedia article on the Enriques-Kodaira classification, before the classification itself, the following sentence appears:
For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or ...

**1**

vote

**2**answers

349 views

### Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles

The Borel--Bott--Weil Theorem gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector ...

**0**

votes

**4**answers

579 views

### Which are the recommended books for an introductory study of complex manifolds? [closed]

Are there any good introductory type of books that is focus on complex manifolds?
Thanks.

**1**

vote

**1**answer

128 views

### Are generically trivial finite unramified morphisms trivial

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.
Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It ...

**8**

votes

**0**answers

427 views

### How much algebraic geometry do I need to study complex geometry?

As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...

**13**

votes

**0**answers

317 views

### Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...

**9**

votes

**1**answer

199 views

### When are the Dolbeault and de Rham dgas homotopy equivalent?

Let $M$ be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on $\Omega_{\mathbb C}^\bullet(M)$ have isomorphic ...

**1**

vote

**1**answer

221 views

### Symplectic and Holomorphic Vector Bundles

As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the ...

**5**

votes

**1**answer

176 views

### A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...

**14**

votes

**3**answers

452 views

### Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$

The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some ...

**22**

votes

**3**answers

890 views

### What are the higher homotopy groups of a K3 suface?

All K3 surfaces have the same homotopy type. What are their higher homotopy groups?
I know that $\pi_1$ is trivial, and $\pi_2$ is $\mathbb{Z}^{22}$.
Even if the answer isn't known in all degrees, ...

**4**

votes

**2**answers

356 views

### What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds?

Given a complex semi-simple Lie group $G$, it acts smoothly on the dual $\frak{g}^*$ of its Lie algebra $\frak{g}$ by the coadjoint action. The orbits of that action are called coadjoint orbits.
A ...

**7**

votes

**1**answer

161 views

### a formula about conic bundles

Let $\pi:Z\to S$ be a conic bundle over a smooth complex surface $S$. I'd like to know how to prove that $-\pi_{*}K_{Z}^{2}=4K_{S}+\Delta$, where $\Delta$ denotes the locus in $S$ over which the ...

**3**

votes

**0**answers

253 views

### Classification of complex Kronecker foliations

Let $\theta \in \mathbb{C}$ be a fixed complex number. The submersion $f:\mathbb{C}^{2}\to \mathbb{C}\; \text{with}\; f(x,y)=y-\theta x$ defines a complex foliation on $\mathbb{C}^{2}$. Consider the ...

**3**

votes

**2**answers

298 views

### How can one determine if a singularity is simple?

Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin.
If we have the following types of ...