**0**

votes

**0**answers

52 views

### Contact and CR Examples

What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR ...

**4**

votes

**0**answers

314 views

### Ravi Vakil: Foundations of Algebraic Geometry, Exercise 18.4 J [closed]

I am reading Vakil's note and do not know how to do the exercise 18.4 J.
Show that the degree of a vector bundle over a regular projective curve over a field $k$ is the degree of its determinant ...

**6**

votes

**1**answer

158 views

### Connectivity of complements of Stein opens

Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a ...

**1**

vote

**0**answers

82 views

### Ohsawa-Takegoshi extension theorem along snc divisor

Let $f:X\setminus D\to Y$ be a holomorphic fibre space and $X,Y$ are Kahler manifolds, where $D$ is a snc divisor on $X$. Can Ohsawa-Takegoshi extension theorem, imply that there exists a holomorphic $...

**8**

votes

**2**answers

189 views

### Inequality on Kähler classes

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let
$\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz
operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. ...

**1**

vote

**0**answers

109 views

### horizontal lift along fibres

Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local
coordinate $(s_1,...,s_d)$ of
$Y$
and a local coordinate $(z_1,...,z_n)$ of ...

**4**

votes

**1**answer

172 views

### Smooth algebraic stacks with precisely two $\mathbb C$-objects

In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...

**5**

votes

**1**answer

275 views

### Is the automorphism group of a Calabi-Yau variety an arithmetic group

Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a ...

**4**

votes

**0**answers

129 views

### Deformations of the moduli space of ppav's

Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor).
Can one compute ...

**2**

votes

**2**answers

378 views

### algebraic leaves of foliation on a product of two curves

Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form $p_1^*(\...

**15**

votes

**1**answer

1k views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**11**

votes

**1**answer

332 views

### Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures

I'm new here. I hope to do it right!
I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations.
Let us take a smooth complex variety $X$ and a ...

**9**

votes

**0**answers

113 views

### Holomorphic vector fields on compact complex manifolds with trivial canonical bundle

Let $M$ be a compact complex manifold whose canonical bundle $K_M$ is holomorphically trivial. Is it possible for $M$ to admit a non-zero holomorphic vector field with zeroes? Equivalently, using a ...

**2**

votes

**0**answers

93 views

### Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...

**9**

votes

**1**answer

826 views

### Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...

**1**

vote

**0**answers

48 views

### moduli space of curves under prescribed tangency conditons

We consider an irreducible component of the Hilbert Scheme of curves in $\mathbb P^2$. Denote it as $\mathcal D.$ We fix a line $L$ and a point $A\in L.$ Denote $\mathcal D_0$ as the subscheme of $\...

**5**

votes

**0**answers

138 views

### The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold

Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on $\Omega^{(0,\bullet)}...

**1**

vote

**2**answers

142 views

### Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition.
Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...

**3**

votes

**0**answers

183 views

### some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus Y$...

**0**

votes

**0**answers

110 views

### self intersection of a curve in a surface

Suppose $S$ is a compact complex surface, $C\subset S$ is a one dimensional irreducible subvariety (a curve). Suppose further, there exists a family of biholomorphism of $S$ nearby the identity map. ...

**7**

votes

**1**answer

217 views

### Fiberwise criterion for a stack to be a gerbe

Let $f:X\to Y$ be a morphism of algebraic stacks.
If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.
I'm wondering about analogues of this fiberwise ...

**8**

votes

**2**answers

250 views

### Example of a non-Kähler manifold with varying plurigenera

Let $X \stackrel{\pi}{\to} \mathbb{D}$ be a proper holomorphic family with fibres $X_t = \pi^{-1}(t)$. Siu proved, when the $X_t$'s are projective, that the plurigenera $h^0(X_t, mK_{X_t})$ are ...

**3**

votes

**1**answer

171 views

### Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian ...

**3**

votes

**0**answers

47 views

### Resources on a smooth topos containing complex analytic/holomorphic geometry

In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry.
First of all: When Urs writes complex analytic geometry, does he mean ...

**8**

votes

**1**answer

439 views

### Intuition for Picard-Lefschetz formula

I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the setup, we ...

**0**

votes

**1**answer

110 views

### generic irreduciblity

Suppose we have a proper morphism $f:X\rightarrow Y$ and $0\in Y$. If the fiber $f^{-1}(0)$ is irreducible and reduced, is the set $\{y\in Y|f^{-1}(y) \text{ is irreducible and reduced}\}$ open?

**0**

votes

**0**answers

52 views

### Cohomology adding the infinite point

Let $F$ be a closed subset of $\mathbb{C}$. (We assume $F \neq \emptyset, F \neq \mathbb{C},$ and $0 \notin F$.)
Of course $F$ is not a closed subset of $\overline{\mathbb{C}}$ in general but it ...

**0**

votes

**1**answer

160 views

### functoriality of hilbert scheme

suppose $f:X\rightarrow Y$ is a morphism between two schemes over scheme $S.$ Do we have the morphism between their hilbert schemes, i.e. is there a natural morphism $Hilb(X/S)\rightarrow Hilb(Y/S)$ ...

**3**

votes

**1**answer

144 views

### can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly?

Can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly? Do you have any idea of which paper has disscussed this topic? For example, what is the ...

**0**

votes

**0**answers

47 views

### dimension of singular set of torsion free sheaves over a unit disc

Suppose $D\subset\mathbb C$ is a unit disc and $\mathcal F$ is a torsion free analytic coherent sheaf over $D$. Define $S(\mathcal F)=\{x\in D|\mathcal F_{x}\, is \,not\, locally\, free\}$. Is $S(\...

**6**

votes

**1**answer

151 views

### Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...

**3**

votes

**1**answer

191 views

### Kobayashi distance function on the upper half-space

I asked this question already in mathstackexchange but got no answer, so I ask it again here.
Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ ...

**2**

votes

**0**answers

185 views

### Deformation of compact complex manifolds

In Kollár's book, Rational curves on Algebraic Varieties, he states the following theorem [II Theorem 1.7].
For a reltative projective flat reduced curve $C$ over an irreducibles base $S$ and a ...

**3**

votes

**0**answers

85 views

### Is there a correspondence between counting curves in P^2 blown up at a point and counting curves in P^2?

Let $X$ be $\mathbb{P}^2$ blownup at one point
and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$
denote the class of a line and the exceptional divisor respectively.
Let $\mathcal{L}...

**2**

votes

**1**answer

158 views

### Division and multiplication that preserve Euclidean norms

I am looking for ways to define
$$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$
where $x,y\in \mathbb{R}^n$ such that
$$\left\|\frac{1}{x}\right\|=\frac{1}{\|...

**13**

votes

**2**answers

459 views

### regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**0**

votes

**0**answers

70 views

### Kahlerness of the projectivized cotangent bundle [duplicate]

Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient ...

**6**

votes

**0**answers

181 views

### Bogomolov-Beauville-Fujiki form, algebraically

Let $M$ be a compact hyperkahler manifold, i.e. a manifold with three
complex structures $I,J,K$ defining an action of quaternions on the
tangent bundle and a metric which is Kahler with respect to ...

**5**

votes

**1**answer

194 views

### Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...

**2**

votes

**0**answers

99 views

### Elliptic fibration arising from a higher genus linear system

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.
Let $L\subset H$ be a ...

**2**

votes

**0**answers

88 views

### How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...

**3**

votes

**1**answer

200 views

### A Bertini-type result for hypersurfaces containing a subvariety

Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it).
Is there a smooth, ...

**0**

votes

**0**answers

81 views

### Spectrum of the Grassmannian Laplacian

The spectrum of the Laplacian (with respect to the Fubini--Study metric) was addressed in this old question. Does anyone know if these results have been extended to the Grassmannians?

**0**

votes

**0**answers

50 views

### Products Laplacian Eigen-Functions over a Kaehler Manifold

I've been trying to learn a little about Laplacians acting on the smooth functions of a compact Kaehler manifold, and made the following (possibly incorrect) observation:
Let $\{f_i\}$ be a set of ...

**7**

votes

**0**answers

134 views

### Topology of family of complex varieties

It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that:
For a proper flat map $f \colon X \rightarrow \Delta$, where
$X$ is a complex ...

**1**

vote

**1**answer

91 views

### What is the Fano index for Hermitian symmetric spaces of compact type?

As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...

**2**

votes

**0**answers

134 views

### Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...

**15**

votes

**3**answers

967 views

### Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

A particle physicist asked me the above question. Let me try to make it more precise. Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension ...

**1**

vote

**1**answer

155 views

### Elliptic fibrations with few singular fibers

It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them.
It is not difficult to prove that an ...

**2**

votes

**1**answer

121 views

### Pencils in very ample linear systems without curve in its base locus

If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...