# Tagged Questions

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votes

**0**answers

79 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
...

**0**

votes

**1**answer

732 views

### direct image of currents

I'm studing currents from Demailly's Complex Geometry, and the author defines the direct image of a current by a $C^{\infty}$ map and also for the case of a submersion. My question is about the ...

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**2**answers

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+400

### 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 this related post and also the following post.
Added : According to their ...

**1**

vote

**1**answer

109 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 $G_2\to U$ ...

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votes

**2**answers

193 views

### Fundamental groups of normal complex quasi-projective varieties

I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?

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**0**answers

56 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 ...

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vote

**1**answer

358 views

### Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as
...

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**0**answers

111 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 ...

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**0**answers

90 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 $X \subset \bar{X}$ ...

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vote

**1**answer

143 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 ...

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88 views

### “Non-symmetric” Polarized Hodge Structure?

In the definition of a polarized Hodge structure (see here for a definition) is the assumption that the generalized Hodge--Riemann pairing is symmetric up to a possible sign. Does anybody work with ...

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**0**answers

83 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 ...

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votes

**2**answers

589 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 ...

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**1**answer

91 views

### Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators.
For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...

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**1**answer

246 views

### A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$.
Question: Is the connectivity of ...

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votes

**1**answer

196 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, ...

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**0**answers

50 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 ...

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**0**answers

195 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 ...

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vote

**0**answers

108 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 ...

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**1**answer

130 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 ...

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**1**answer

100 views

### Singularities of the Remmert reduction of a holomorphic convex manifold

Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...

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**2**answers

226 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 ...

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**0**answers

69 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 ...

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**1**answer

208 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 ...

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votes

**3**answers

351 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 ...

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**1**answer

186 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 $t \in D \backslash \{0\}$, ...

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**0**answers

40 views

### Lifting $SHFC$ to non singular foliations

In this question we would like to complexify the idea in the following post.
We would like to lift a "singular holomorphic foliation by curves", briefly "SHFC", of $\mathbb{C}P^{2}$ to a ...

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votes

**1**answer

95 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}} ...

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62 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 ...

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104 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 ...

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**0**answers

187 views

### Classification of the Kähler Structures on the Sphere

Is there a classification of the Kähler structure on the sphere? More generally, is there a classification of the Kähler structures on the complex projective spaces? Even more generally, what about ...

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votes

**1**answer

202 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 ...

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**1**answer

548 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 ...

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votes

**1**answer

361 views

### isolated points

can $E_c(T)=\{x\in X~:~\nu(T,x)\geq c\}$ have isolated point?
where T is a positive current of bidegree 1, c is a positive real number, $X$ is a complex variety, and $\nu(T,x)$ is the Lelong number of ...

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**5**answers

2k views

### Dolbeault cohomology of Hopf manifolds

This should be straightforward; I'm sorry if it's too much so. Can someone point me to a reference which computes the Dolbeault cohomology of the Hopf manifolds?
Motivation: I'd like to work ...

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vote

**1**answer

121 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?

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votes

**1**answer

380 views

### Why does the Lefschetz Operator not Square to Zero? [closed]

I'm trying to learn about the Lefschetz decomposition but am having a very basic problem: For the fundamental form $K$ of a Kahler metric on a complex manifold $M$, the corresponding Lefschetz ...

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**1**answer

301 views

### ${\bar{\partial}}$-geometrically formal ?

A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler compact complex ...

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**0**answers

92 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 ...

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**0**answers

73 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
...

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**0**answers

180 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 ...

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**1**answer

232 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 ...

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**0**answers

245 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 ...

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votes

**1**answer

117 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 ...

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vote

**2**answers

241 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 ...

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votes

**4**answers

316 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.

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votes

**3**answers

286 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 ...

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**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 ...

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**1**answer

121 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 ...

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votes

**0**answers

333 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. ...