Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

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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 ...
Christian's user avatar
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3 votes
2 answers
554 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^*(\...
Lya's user avatar
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18 votes
2 answers
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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?
user312955's user avatar
15 votes
1 answer
1k 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 ...
a_g's user avatar
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9 votes
0 answers
473 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 ...
Joey's user avatar
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2 votes
0 answers
280 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 ...
Alex M.'s user avatar
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11 votes
1 answer
1k 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 $...
Amir Baghban's user avatar
1 vote
0 answers
50 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 $\...
user42804's user avatar
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6 votes
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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)}...
Pavel Katzo's user avatar
1 vote
2 answers
596 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 ...
asv's user avatar
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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, $X^*=X\setminus Y$...
Feng Hao's user avatar
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0 answers
519 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. ...
user42804's user avatar
  • 1,091
7 votes
1 answer
368 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 ...
Marci Senf's user avatar
7 votes
2 answers
394 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 ...
msteve's user avatar
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3 votes
1 answer
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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 ...
Marci Senf's user avatar
5 votes
0 answers
151 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 ...
Georg Lehner's user avatar
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27 votes
1 answer
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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 ...
user84144's user avatar
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0 votes
1 answer
134 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?
user42804's user avatar
  • 1,091
0 votes
1 answer
342 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 $Hilb(X/S)\rightarrow Hilb(Y/S)$ ...
user42804's user avatar
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4 votes
1 answer
249 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 ...
user42804's user avatar
  • 1,091
0 votes
0 answers
121 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(\...
user42804's user avatar
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6 votes
1 answer
391 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 $\rho$ ...
Entaou's user avatar
  • 285
3 votes
1 answer
384 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 $g\times g$ ...
Jack's user avatar
  • 637
3 votes
0 answers
274 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 ...
Chieh LIU's user avatar
  • 147
3 votes
0 answers
100 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}...
Ritwik's user avatar
  • 3,235
2 votes
1 answer
277 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}{\|...
Ma Na's user avatar
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14 votes
2 answers
532 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 ...
Ma Na's user avatar
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0 votes
0 answers
113 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 ...
asv's user avatar
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8 votes
0 answers
1k 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 ...
Dima Sustretov's user avatar
4 votes
1 answer
453 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 ...
physicsoutsideborders's user avatar
3 votes
0 answers
172 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 ...
Alan Muniz's user avatar
4 votes
0 answers
168 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 ...
Han Jin Ma's user avatar
4 votes
1 answer
478 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, ...
Li Yutong's user avatar
  • 3,362
8 votes
0 answers
224 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 algebraic ...
user84144's user avatar
  • 2,769
1 vote
1 answer
182 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 ...
user42804's user avatar
  • 1,091
2 votes
0 answers
166 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 ...
Alon's user avatar
  • 75
15 votes
3 answers
1k 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 ...
John Baez's user avatar
  • 21.3k
3 votes
1 answer
918 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 ...
Alan Muniz's user avatar
2 votes
1 answer
300 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 ...
Alan Muniz's user avatar
9 votes
1 answer
451 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 ...
Julien Grivaux's user avatar
3 votes
1 answer
160 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 ...
Andrew McHugh's user avatar
3 votes
1 answer
483 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} ...
jorst's user avatar
  • 359
4 votes
0 answers
245 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 ...
Mike Shulman's user avatar
1 vote
0 answers
216 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 ...
asv's user avatar
  • 21.1k
2 votes
0 answers
172 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 ...
gsvr's user avatar
  • 235
4 votes
0 answers
207 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 ?
user42804's user avatar
  • 1,091
1 vote
0 answers
169 views

Line bundles with vanishing cohomology on Calabi-Yau manifold

Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$. Is anything known about such ...
Mathemage's user avatar
1 vote
0 answers
202 views

Stability of automorphism group of complex manifolds

Are there any stability theorems of analytic automorphism groups concerning the deformation of complex manifolds. For example, in the case of K3 or Calabi-Yau.
Nobu's user avatar
  • 79
12 votes
0 answers
149 views

Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
Quidam_t's user avatar
  • 121
2 votes
1 answer
193 views

Control of a meromorphic function according to distance between its zeros

My question is rather philosophical : can a meromorphic function of normalized norm with simple zeros on the flat torus stay close to zero on a large set when its zeros are far from each other ? The ...
Fougeroc's user avatar

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