**3**

votes

**1**answer

153 views

### holomorphic sectional curvature and total scalar curvature

In a paper of Heier and Wong, It is written that from a pointwise argument due to Berger does follow that the scalar curvature (and thus also the total scalar curvature) of a Kaehler metric of ...

**4**

votes

**2**answers

405 views

### When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?

Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (non-compact) complex manifold $X$. If $X$ and $Y$ are two (non-compact) complex manifolds, then $\mathrm{Aut}(X)$ and ...

**1**

vote

**1**answer

78 views

### Removing a hyperplane from flag manifolds

It should be known that if we remove a compact complex codimension one submanifold $X$ (hyperplane) of a flag manifold $Z=G/P$, then $Z\setminus X$ is a Stein manifold.
I was wondering if anyone can ...

**8**

votes

**2**answers

304 views

### Uniformization of Kodaira fibered surfaces

Consider a Kodaira fibration. i.e. a smooth non-isotrivial fibration $X\rightarrow C$ with $X$ a smooth complex surface and $C$ a smooth complex curve, such that both the genus of $C$ and genus of the ...

**9**

votes

**1**answer

519 views

### There are only finitely many varieties up to deformation

Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...

**7**

votes

**1**answer

344 views

### Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...

**2**

votes

**1**answer

135 views

### Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain

Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?

**3**

votes

**2**answers

365 views

### Is a holomorphic family whose fibers are all smooth locally trivial?

Let $\pi : X \to B$ be a proper, surjective holomorphic submersion, where both $X$ and $B$ are compact Kahler manifolds. Assume that all the fibers $X_b = \pi^{-1}(b)$ are smooth. Is the family $\pi : ...

**3**

votes

**1**answer

133 views

### Question about a variant of the index of a Fano manifold

Given a smooth Fano variety $X$ over $\mathbb{C}$, we can define the index, $I(X)$, as the divisibility of $-K_X$ inside of $Pic(X)$. There is a theorem which states that $I(X) \leq n+1$, where $n$ is ...

**2**

votes

**0**answers

112 views

### Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular).
Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that
...

**4**

votes

**1**answer

313 views

### Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...

**6**

votes

**2**answers

530 views

### Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, ...

**0**

votes

**1**answer

196 views

### question about the developing map

I'm having some trouble finding literature on the developing map.
All the sources I could find on it seem to refer to thurston's definition in either:
http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf
or
...

**1**

vote

**1**answer

247 views

### The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...

**6**

votes

**3**answers

299 views

### Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...

**4**

votes

**1**answer

266 views

### If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$

I'm trying to prove the following:
Let $L$ be a positive holomorphic line bundle on a compact complex manifold $X$. For any hermitian holomorphic vector bundle $E$ on $X$, there is $k \in ...

**0**

votes

**1**answer

288 views

### A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.

**0**

votes

**2**answers

274 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?

**2**

votes

**1**answer

177 views

### Orbits of an action

How can I explicitly calculate all the orbits of the action of $SO(3)$ on $\mathbb C\mathbb P_2$?
For example I know that one of the orbits is the quadric $\{[z_0:z_1:z_2]\in\mathbb C\mathbb P_2: ...

**2**

votes

**1**answer

124 views

### Existence of logarithmic structures and d-semistability

I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a ...

**1**

vote

**1**answer

212 views

### a question on the space of divisors on a curve

Let $X$ a complex curve and $x\in X$ a point.
We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group.
...

**3**

votes

**2**answers

219 views

### Ramification divisor and degenerate locus of jacobian

Let $f : X \to Y$ be a finite morphism between compact complex manifolds of the same dimension $n$. We denote by $R_f \subset X$ the ramification divisor of $f$ and by $J_f \subset X$ the set of ...

**0**

votes

**0**answers

126 views

### Dimension of $H^{0}(X,TX)$ with $X$ resolution of $\mathbb{C}^{m}/G$

Let $m\geq 2$ and $G$ a finite subgroup of $U(m)$ acting freely on $S^{2m-1}\subset\mathbb{C}^{m}$. Let $X$ be a resolution of $\mathbb{C}^{m}/G$ i.e. $X$ is a smooth complex manifold with a map
...

**3**

votes

**2**answers

308 views

### How to explicitly see the ramification over infinity

Take the equation $y^{d}=\Pi_{1}^{n}(x-t_{i})^{m_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum ...

**0**

votes

**0**answers

81 views

### An integral over Clifford torus

This is part of my research project and I hope somebody can help with it.
Let $F_{ij}$ be the antisymmetric function
$$\frac{1}{4}L^4 \left(\text{cos}\left[\frac{f_{2j}}{2}-\frac{g_{2j}}{2}\right] ...

**1**

vote

**0**answers

59 views

### a regular type of U-rank > 1 in the theory of compact complex spaces

What is an example, in the theory of compact complex spaces, of a regular (i.e. orthogonal to all its forking extensions) type which is of U-rank strictly greater than 1?
update: after witnessing ...

**8**

votes

**1**answer

312 views

### Does there always exist a line bundle whose Chern class represents an integer symplectic form?

Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, ...

**6**

votes

**3**answers

400 views

### Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$

Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...

**6**

votes

**2**answers

309 views

### Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$

This question is related to this MO question and this MSE question.
Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes ...

**2**

votes

**1**answer

347 views

### Question about a Lefschetz hyperplane type theorem

Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample.
I ...

**2**

votes

**0**answers

95 views

### topology generated by irreducible componets of $\Gamma$-invariant closed sets

For an analytic space $U$ equipped with an action of a group $\Gamma$,
call a subset $Z\subseteq U$ $\Gamma$-closed iff
it is a closed analytic subset and each of its irreducible components
is an ...

**4**

votes

**1**answer

156 views

### Existence of nodal curves in a linear system

Let $S$ be a projective surface and $L$ an ample line bundle on $S$. The Severi variety $\mathcal V_{\mathcal L,\delta}$ parametrizes curves with $\delta$ nodes and no other singularities in the ...

**8**

votes

**1**answer

277 views

### Can one use Brownian motion to prove that two manifolds are not conformally equivalent?

Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a ...

**5**

votes

**3**answers

368 views

### $E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$

I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
$E$ is a holomorphic vector bundle.
There is a Dolbeault operator ...

**3**

votes

**1**answer

197 views

### Exact sequence for divisor class groups

Let $X$ be a either a projective scheme or a compact complex space. Then one has an exact sequence $$ (1) \quad 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} ...

**0**

votes

**0**answers

33 views

### Covers homotopic to Leray covers

Not well-versed in algebraic geometry (my research is in other things), so apologies in advance if my terminology is lazy and/or wrong.
I'm trying to do some Cech cohomology on $\mathbb{C}P^3$ and ...

**3**

votes

**1**answer

224 views

### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...

**2**

votes

**1**answer

163 views

### What happens to small squares in Riemann mapping?

I have a square S, and I want to convert it to the unit disc D.
The Riemann mapping theorem says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
...

**3**

votes

**0**answers

104 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 $g$ ...

**2**

votes

**0**answers

204 views

### Least area minimal hypersurface of $\mathbb C P^{n+1}$

After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...

**10**

votes

**1**answer

271 views

### Which complex manifolds embed into tori?

If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to ...

**0**

votes

**1**answer

260 views

### A question about a two form and a $(1,1)$ form on a compact Kähler manifold

Suppose $\omega$ is a real closed $(1,1)$ form on a compact Kähler manifold. If we have a real $d$-closed two form $\sigma$ such that $[\sigma]=[\omega] \in H^2(M)$, can we claim that this two form ...

**12**

votes

**3**answers

440 views

### Three-dimensional compact Kähler manifolds

Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.
$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a ...

**1**

vote

**3**answers

379 views

### prequantization on $TM \bigoplus T^*M$

Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin ...

**0**

votes

**1**answer

117 views

### Why can't hyper-kahler manifolds have a connection with torsion?

I have often seen the statement that Hyper-Kahler (HK) manifolds have torsion-free connections. In general relativity, however, one is usually taught that the connection is something that you can ...

**3**

votes

**2**answers

390 views

### What is the closure of space of polynomials in a dense subspace along with a marked point equal to?

EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots ...

**0**

votes

**1**answer

98 views

### Is the space of degree $d$ curves with marked smooth points dense inside the space of curves with marked points?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero
homogeneous degree $d$ polynomials in three variables upto scaling, where
$\delta_d = \frac{d(d+3)}{2} $
(basically degree ...

**3**

votes

**1**answer

176 views

### General position argument for reasonable spaces

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of non zero homogeneous degree $d$ polynomials in three variables up to scaling, where
$\delta_d:= \frac{d(d+3)}{2} $. Given a point $p ...

**4**

votes

**3**answers

505 views

### What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset ...

**4**

votes

**2**answers

261 views

### Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...