**2**

votes

**0**answers

55 views

### Group of real analytic isometries of $g$-fold product of the Poincare upper half plane

Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...

**7**

votes

**2**answers

481 views

### Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...

**3**

votes

**1**answer

141 views

### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...

**0**

votes

**0**answers

48 views

### What is an induced Abel Jacobi map?

Let $X$ be a compact Kahler threefold, $C\subset X$ be a smooth curve, $\tau\colon\tilde{X}\to X$ be the blow up of $X$ along $C$, $j\colon E\to \tilde{X}$ be rthe exceptional divisor, $\tau_E$ be the ...

**3**

votes

**2**answers

344 views

### Three and a half basic questions on the Weil restriction of scalars

(This is reposted from mathstackexchange, where it received no answer so far.)
I am currently trying to get familiar with the Weil Restriction functor.
For a finite field extension $L|K$ it ...

**7**

votes

**1**answer

261 views

### Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes

Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ ...

**1**

vote

**1**answer

117 views

### extension of holomorphic mappings

In 1971 Phillip.A.Griffith proved that
Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact ...

**3**

votes

**1**answer

137 views

### Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...

**1**

vote

**0**answers

40 views

### Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = ...

**14**

votes

**3**answers

2k views

### Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when ...

**4**

votes

**0**answers

147 views

### Explicit metrics on non-compact Calabi-Yau threefolds

I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known.
For instance, an important class of such spaces can be constructed algebraically, including local ...

**1**

vote

**2**answers

154 views

### Transformations that leave the Plucker embedding of G(2,4) invariant

I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by ...

**4**

votes

**0**answers

70 views

### Metrics on Teichmüller spaces

I know that Teichmüller $\mathcal{T}_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other ...

**3**

votes

**0**answers

86 views

### Toponogov comparison theorem for complex manifold

I would like to know some reference for the Toponogov comparison theorem for complex manifolds, in particular for complex manifolds with bounded holomorphic sectional curvature. As far as I know, the ...

**0**

votes

**0**answers

114 views

### The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...

**0**

votes

**1**answer

118 views

### How does $H_1$ change after projection

Suppose $X\subset \mathbb{CP}^N$ is a $n$ dimensional projective manifold (and complex dimension $n>1$), take a general projection $p\colon X\to\mathbb{CP}^{n+1}$. Suppose $H_1(X)$ is nontravial. ...

**1**

vote

**1**answer

234 views

### Two limit cycles which lie on the same leaf

Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

**4**

votes

**1**answer

192 views

### Infinitesimal deformations of fake projective planes (or ball quotients)

This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...

**6**

votes

**1**answer

416 views

### Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry.
Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature?
I googled ...

**1**

vote

**1**answer

156 views

### Orientation form on the blow up of a Kaehler manifold

Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...

**5**

votes

**1**answer

239 views

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

**59**

votes

**2**answers

4k views

### Complex structure on $S^6$ gets published in Journ. Math. Phys

A paper by Gabor Etesi was published that purports to solve a major outstanding problem:
Complex structure on the six dimensional sphere from a spontaneous symmetry breaking
Journ. Math. Phys. 56, ...

**1**

vote

**0**answers

88 views

### Looking for examples of holomorphic maps to $\mathbb{P}^1$ with certain property

I would like to know any example of nonconstant holomorphic map $f:X\to\mathbb{P}^1$ such that $K_X\cong f^*\mathcal{O}(2n)$ for some positive integer $n$, where $K_X$ is the canonical bundle of $X$.
...

**2**

votes

**0**answers

101 views

### Do smooth manifolds create colimits for complex manifolds?

Suppose we have a diagram $D$ in the category $\textrm{Diff}_\mathbb{C}$ of complex manifolds, and suppose this diagram has a colimit $L$ after inclusion in the category $\textrm{Diff}$ of smooth ...

**4**

votes

**1**answer

358 views

### Tate twist and comparison between Betti and de Rham cohomology

In Deligne's paper "Hodge cycles on abelian varieties" (see page 11 of http://jmilne.org/math/Documents/Deligne82.pdf) he says that the following diagram fails to commute by a factor of $(2 \pi i)^m$, ...

**0**

votes

**0**answers

131 views

### Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...

**0**

votes

**1**answer

187 views

### Almost complex structure and nontrivial idempotents

Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...

**4**

votes

**1**answer

211 views

### Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't ...

**1**

vote

**0**answers

142 views

### Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold

Let $(M,\omega)$ be a Kahler manifold with Kahler integral two-form $\omega$ and let $(L,h)$ be a rank-one complex vector bundle over $M$ equipped with a fixed hermitian metric $h$. I am interested in ...

**2**

votes

**0**answers

417 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics ...

**2**

votes

**0**answers

92 views

### How to study Kähler metrics singular along a submanifold of codim 2?

Let $M$ be a compact complex manifold, $S\subset M$ a submanifold of codimension $2$, let $\omega$ be a k\"ahler metric on $M\setminus S$. Then we know by Reese Harvey's paper "Removable singularities ...

**4**

votes

**1**answer

506 views

### Topology of algebraic varieties

Let $X$ be a projective variety (lets say normal and irreducible) with the topology coming from being a subspace of $\mathbb{P}^N$ (and not the Zariski topology). Surely one can then define the ...

**5**

votes

**1**answer

290 views

### Algebraic spaces which are automatically schemes

Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.
If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...

**6**

votes

**1**answer

231 views

### Pontryagin Forms and Special Holonomy

Let $(M,g)$ be a Riemannian manifold. Recall that the $k^{th}$-Pontryagin class is a topological invariant which, by classical Chern-Weil theory, can be represented using the so-called Pontryagin ...

**0**

votes

**0**answers

106 views

### Geometric transitions of Calabi-Yau threefolds

Let me start with the definition of geometric transition:
Let $Y$ be a Calabi–Yau 3–fold and $\phi: Y \rightarrow \overline Y$ be a birational contraction onto a normal variety. If there exists a ...

**2**

votes

**1**answer

155 views

### Constant spinors from constant forms

Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S_{\mathbb{C}}$ its complex spinor bundle. Then:
$S_{\mathbb{C}}\simeq \Lambda_{\mathbb{C}}(X)$
Let $\nabla$ ...

**0**

votes

**2**answers

182 views

### Special connection of vector bundle over real manifold

Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...

**6**

votes

**1**answer

299 views

### Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**1**answer

231 views

### What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**0**answers

131 views

### Varieties acted upon faithfully by an abelian variety

Let $X$ be a smooth projective variety over the complex numbers. Suppose that some positive-dimensional abelian variety $A$ acts faithfully on $X$.
Examples of such varieties $X$ are provided by ...

**4**

votes

**0**answers

288 views

### What's the name of this branched covering?

I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated ...

**2**

votes

**1**answer

191 views

### Can a rigid CY threefold have infinitely many automorphisms

Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?
N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) ...

**2**

votes

**2**answers

209 views

### Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...

**2**

votes

**0**answers

71 views

### Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I ...

**7**

votes

**0**answers

90 views

### Equation of the curve corresponding to a polarization of an abelian surface

Let $\mathbb{C}^2/\Lambda$ be a polarized abelian surface. I think it is well-known how to write down the equation of the divisor corresponding to the polarization, in terms of theta functions etc. ...

**2**

votes

**0**answers

68 views

### What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...

**0**

votes

**0**answers

164 views

### Are del-Pezzo surfaces complete intersections?

Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$
to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$
for some $n$.
$\textbf{Question:}$ Can $X_k$ ...

**2**

votes

**0**answers

77 views

### Does the monodromy of such VHS have to be trivial

Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...

**1**

vote

**0**answers

112 views

### Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...

**3**

votes

**1**answer

112 views

### Is every positive $(n-1,n-1)$ form almost decomposable?

Suppose $(X,\omega)$ is a compact hermitian manifold of complex dimension $n$, and $\alpha$ is a smooth $(n-1,n-1)$ positive form, i.e., $\alpha = a \omega ^{n-1} + \displaystyle \sum _{i=1} ^m ...