**1**

vote

**1**answer

76 views

### Triviality of a circle fibration induced by an almost complex structure

Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$
$J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More ...

**-2**

votes

**1**answer

155 views

### Is it easy to see that a cubic surface $V$ in $CP^3$ has no holomorphic 2-forms? [closed]

More specifically, what facts do you need to know to conclude $H^2(V) = H^{1,1}$? In general, are there hypersurfaces in $CP^n$ without holomorphic $k$-forms for some $k$?

**4**

votes

**0**answers

242 views

### Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds)
And Ricci curvature of Narasimhan-Simha ...

**0**

votes

**1**answer

204 views

### Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:
https://en.wikipedia.org/wiki/...

**2**

votes

**0**answers

133 views

### Question regarding a lemma in Principles of Algebraic Geometry

My question is regarding the lemma on page 81 of the book by Griffiths and Harris.
The lemma says the following:
A $\bar{\partial}$-closed form $\psi\in Z^{p,q}_{\bar{\partial}}(M)$ is of minimal ...

**3**

votes

**0**answers

61 views

### What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...

**7**

votes

**1**answer

241 views

### Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:
For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...

**10**

votes

**3**answers

548 views

### What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...

**2**

votes

**1**answer

303 views

### A conjecture from Jean Varouchas on Kahler varieties

Conjecture: Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces.
If $X$ is Kahler, is $X'$ Kahler?
This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from ...

**3**

votes

**0**answers

258 views

### Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that
$$\frac{\partial \omega(t)}{\partial t}=-Ric_{X/Y}(\omega(t))-\omega(...

**6**

votes

**0**answers

138 views

### Deformation of Complex Spaces

I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology.
Is there any other modern reference to this ...

**2**

votes

**0**answers

415 views

### On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) \subseteq TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. Let $\bar{J} : TTM \longrightarrow ...

**1**

vote

**0**answers

66 views

### Complex Structure Moduli of Elliptic Fibrations

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.
I know how it is done for the generic Weierstrass ...

**0**

votes

**0**answers

53 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

318 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

162 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

85 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

197 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

110 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

175 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

283 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

130 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

379 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^*(\...

**16**

votes

**2**answers

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

340 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

118 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

96 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

835 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

141 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

147 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

185 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

220 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

251 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

48 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

475 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

161 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

146 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

155 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

192 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

187 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

86 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

463 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 ...