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.
3,129
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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 ...
3
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2
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554
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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^*(\...
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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?
15
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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
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473
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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
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280
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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 ...
11
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1
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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 $...
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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 $\...
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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)}...
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2
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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 ...
4
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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, $X^*=X\setminus Y$...
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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. ...
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1
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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 ...
7
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2
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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
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1
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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 ...
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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 ...
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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 ...
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1
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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?
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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 $Hilb(X/S)\rightarrow Hilb(Y/S)$ ...
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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 ...
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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
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1
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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
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1
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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 $g\times g$ ...
3
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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
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0
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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
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1
answer
277
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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
$$\left\|\frac{1}{x}\right\|=\frac{1}{\|...
14
votes
2
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532
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regular polygon question
Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...
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113
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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 ...
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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 ...
4
votes
1
answer
453
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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 ...
3
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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 ...
4
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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 ...
4
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1
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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, ...
8
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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 ...
1
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1
answer
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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
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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
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3
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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 ...
3
votes
1
answer
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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
300
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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
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1
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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
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1
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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
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1
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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}
...
4
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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 ...
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216
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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
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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 ...
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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 ?
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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 ...
1
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0
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202
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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.
12
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149
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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 ...
2
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1
answer
193
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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 ...