All Questions
Tagged with complex-geometry moduli-spaces
61 questions
1
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98
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Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
- 111
1
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1
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279
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Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
- 331
1
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0
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51
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Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
- 285
0
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1
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160
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Teichmüller theory for open surfaces?
I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?
My motivation basically is that I would like to find out more about the "...
- 7,127
2
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0
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70
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Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles
Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...
- 223
2
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1
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290
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On the stack of semistable curves
This is a question related to
Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...
- 494
3
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0
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269
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On the normal crossing divisor of $\overline{\mathcal M}_g$
Let $g\geq 2$ be an integer. Let $\overline{\mathcal M}_g$ denote the DM stack of stable curves of genus $g$. It is well-known that the moduli stack is smooth and has a natural normal-crossing divisor ...
- 494
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224
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The genus of hyperplane sections
Let $S$ be a connected smooth projective surface over $\mathbb{C}$.
Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...
- 519
1
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1
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306
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Quiver varieties associated to D_4
Let $Q=(I,\Omega)$ be the $D_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\...
- 1,061
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2
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568
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Explicit example de Rham moduli space of connections
Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers. We can consider different moduli spaces related to those. On one hand we have:
-$M_{Dol}$ the moduli space of stable ...
- 1,061
3
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1
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282
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Integral locus of Hitchin morphism
Let $\Sigma$ be a Riemann surface of genus $g$. To it, we can associated $M_{Dol}$ be the Higgs moduli space of rank $n$ and degree $d$. Fo simplicity let us take $(n,d)=1$. This quasiprojective ...
- 1,061
3
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1
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348
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Non Abelian Hodge theory: underlying structure holomorphic vector bundles
Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,...
- 1,061
3
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0
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233
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Local existence of (quasi)-universal family of sheaves
Let $p : X \to S$ be a projective morphism between two Noetherian $\mathbb{C}$-schemes of finite type with connected fibres. Let $O_X(1)$ be a very ample line bundle on $X$ relative to $S$. Given a ...
- 196
1
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1
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148
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Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map
Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.
Q1. How many such threefolds exist, and ...
- 137
1
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0
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89
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The dimension of parameter space of unstable Higgs bundle
Let $X$:smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$, $\mathcal{M}(r,d)$:moduli space of stable Higgs bundles of rank $r\geq 2$ and degree $d$ on $X$, and $N$:moduli space of stable ...
- 297
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361
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On definition of stable vector/Higgs bundle
Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
- 1,170
4
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0
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109
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Holomorphic maps on moduli space and Deformation theory
Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map
$$f:\mathcal{M}\rightarrow \mathcal{F}$$
means that for each ...
- 4,418
1
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0
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203
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Holomorphic map from a punctured affine line to $M_g$ with Zariski-dense image
Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. Is there a holomorphic map $U\to M_g$ with Zariski-dense image where $U$ is the affine line with ...
user145520
1
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1
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185
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Is there an algorithm on an infinite time Turing machine to compute a dense subset of $M_g$ for large $g$?
Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. If $g$ is a random large integer (e.g. $g=100$) does there exist an algorithm on an infinite time ...
user145520
1
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0
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96
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Geometry of the complex Gauge group
Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$.
Is there a way to endow $\...
- 789
5
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1
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827
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Coarse moduli space versus Kuranishi family
We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...
- 527
4
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0
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117
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Complexifed Gauge action on determinant line bundle and change of metric
Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
- 789
1
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0
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165
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SL(2,R) invariant which are not SL(2,C) invariants
Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...
- 1,435
2
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1
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216
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Is the stack of stable curves with no rational component algebraic?
Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$.
Let $\mathcal{M}_g^{nr}$ be the substack of ...
- 21
6
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1
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769
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Rationality of the moduli space of genus g curves
I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
- 5,760
7
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0
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499
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Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures
Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
user21574
13
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3
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1k
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DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons
As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation ...
- 2,818
4
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0
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254
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Deformation space and Kodaira-Spencer map of cyclic Galois coverings
This question concerns a statement from the paper by Ben Moonen "Special subvarieties arising from families of cyclic covers of the projective line. Documenta Math. 15 (2010)", Lemma 5.5. ii).
More ...
- 2,221
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338
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GAGA for moduli problems
In algebraic geometry moduli problems are described by a functor $F:\mathrm{Sch}^{\mathrm{op}}\to \mathrm{Set}$ and it is clear what a solution to a moduli problem is, namely a scheme X such that $F\...
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272
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Is the analytification of the coarse space equal to the coarse moduli space of the analytification?
If $X$ is a smooth finite type separated DM algebraic stack over $\mathbb C$ with coarse space $X^c$, then do we know whether the analytification of $X^c$ is the coarse space of the analytification of ...
- 113
6
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2
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495
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Riemann Theta Function On Hyperbolic Riemann Surfaces
The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by
$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...
- 989
2
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0
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247
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Moving in the Hurwitz Space?
To my very limited understanding, a Hurwitz space parameterizes branched coverings $(\Sigma,f)$ with a set of given branching data. Here $f:\Sigma\to\mathbb{CP}^1$ is a nonconstant holomorphic map ...
- 783
1
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523
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Quasi-projectivity of the moduli space of Kahler-Einstein Fano varities and vanishing Lelong number
Chi Li, Xiaowei Wang, Chenyang Xu proved the Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds. My question is about when central fibre $X_0$ along Kahler-Einstein Fano ...
user21574
4
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0
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332
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On dimension of the moduli space of abelian differentials on Riemann surfaces
I fear I'm missing something important here, so forgive me if my question is stupid.
Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli ...
- 109
2
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0
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94
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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 ...
2
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0
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146
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Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?
Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...
- 3,245
4
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1
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199
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Singularities of the moduli stack of polarized hyperkahler varieties
Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...
- 41
9
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1
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594
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Singularities of the moduli stack of Calabi-Yau threefolds
Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?
In many cases I ...
- 93
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291
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Properties of finite quotients of quasi-projective varieties
Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...
- 31
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0
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580
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On the Hitchin fibration
I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
- 111
15
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1
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3k
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Kodaira-Spencer theory of deformation done right
I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...
- 2,227
11
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2
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759
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Families of Fano varieties over non-hyperbolic curves
Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be ...
- 9,497
1
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0
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153
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some intuition about the degree of a map
Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...
- 533
2
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0
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115
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Characterisation of convergence in Deligne-Mumford compactifiaction
1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...
- 2,696
9
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1
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713
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There are only finitely many varieties up to deformation
Let $h$ be a polynomial. Then results of several authors (including Chow, Grothendieck, Matsusaka, Mumford, Kollar and Viehweg) imply that the moduli space of polarized varieties with Hilbert ...
- 93
2
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1
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902
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There are many varieties with ample canonical bundle
Let $X$ be a smooth projective connected complex algebraic variety with ample canonical bundle. Let $h$ be the hilbert polynomial of the canonical bundle.
Why is the moduli stack of canonically ...
11
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2
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977
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Proving that a generic variety with ample canonical bundle has no automorphisms
Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivial, see for instance
...
- 246
1
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2
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485
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Isotrivial K3 family and Picard number
Is it true that any family of K3 surfaces over $\mathbb{C}$ whose Picard number is constant is isotrivial? Here isotrivial means locally analytically trivial.
Speculation: Let $\mathcal{M}$ be the ...
- 11
20
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3
answers
2k
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What is the DGLA controlling the deformation theory of a complex submanifold?
Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. ...
- 361
0
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1
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118
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Regular (or complex analytic) functions on M_3
Let $M_3$ be the moduli space of genus three curves over $\mathbb C$.
Are there non-constant regular functions of this space? What about complex analytic functions?
This question is prompted by the ...
- 14.3k