**7**

votes

**0**answers

44 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

59 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 ...

**6**

votes

**1**answer

209 views

### A solution for this equation with a certain condition

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition
\begin{equation}
J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\
...

**1**

vote

**0**answers

44 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 ...

**2**

votes

**0**answers

140 views

### Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor on Kahler variety $X$. I am looking for a proof that such moduli space exists?
The log ...

**4**

votes

**1**answer

481 views

### Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...

**-1**

votes

**0**answers

28 views

### integral curve of a holomorphic vector field on a complex manifold [closed]

Let $V$ be a holomorphic vector field on a complex manifold $M$. An integral curve of $V$ through a point $p\in M$ is a complex curve $\gamma: D\to M$, where $D$ is a connected neighborhood of $0$ in ...

**5**

votes

**0**answers

114 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 ...

**1**

vote

**2**answers

112 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

152 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 ...

**0**

votes

**0**answers

105 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

204 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 ...

**-1**

votes

**0**answers

61 views

### linearly equivalence of line bundles

Suppose we have a complex manifold $X$ with a fmaily of nontrivial holomorphic biholomorphisms $F(t,x)$ where $t\in\mathbb C,|t|<1$. And $F(0,x)=Id$ and $F(t,x)$is holomorphic in all variables. ...

**8**

votes

**2**answers

224 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 ...

**-1**

votes

**0**answers

44 views

### Order of Contact for a general tangent line of a cubic threefold [migrated]

I am trying to solve Exercise 18.21 of Harris "Algebraic Geometry".
In the proof of the unirationality of a smooth cubic threefold X he claims that a general tangent line to X at a general point p ...

**3**

votes

**1**answer

154 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

40 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 ...

**7**

votes

**1**answer

266 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

108 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

50 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

154 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

136 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

43 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 ...

**6**

votes

**1**answer

134 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

176 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

173 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

79 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 ...

**2**

votes

**1**answer

143 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
...

**7**

votes

**2**answers

309 views

### regular polygon and constant potential function

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

67 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

137 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

181 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

96 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

81 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

183 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

72 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

43 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

127 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

80 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

98 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

938 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

126 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

116 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 ...

**9**

votes

**1**answer

273 views

### Cohomology of vector bundles in families

Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat ...

**3**

votes

**1**answer

83 views

### Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact?

Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily ...

**3**

votes

**1**answer

235 views

### Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition
\begin{equation}
...

**3**

votes

**0**answers

87 views

### Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...

**1**

vote

**0**answers

117 views

### Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...

**2**

votes

**0**answers

86 views

### Name for the variety of preimages of a finite morphism

If $f:X\to Y$ is a finite morphism of degree $d$ between two varieties, you get a closed subset of the symmetric product $X^{(d)}$ (or perhaps rather the Hilbert scheme $X^{[d]}$), defined as the ...

**4**

votes

**0**answers

101 views

### classification of homogenous complex manifolds

Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?