All Questions
Tagged with moduli-spaces dg.differential-geometry
52 questions
57
votes
7
answers
8k
views
Maryam Mirzakhani's works
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...
user68866
44
votes
2
answers
5k
views
Meaning/origin of Seiberg-Witten equations/invariants
Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take ...
- 17.5k
23
votes
1
answer
2k
views
Do hyperKahler manifolds live in quaternionic-Kahler families?
A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO.
I'm aware that there are a number of Torelli type theorems now proven for ...
- 13.3k
20
votes
3
answers
2k
views
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
13
votes
3
answers
1k
views
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 ...
- 2,816
13
votes
1
answer
564
views
What is the second fundamental form of moduli space?
Away from the hyperelliptic locus, the moduli of curves immerses
in the moduli of principally polarized abelian varieties. The
ambient space has a riemannian metric, so one can ask about the
second ...
- 8,717
11
votes
1
answer
2k
views
Seiberg-Witten theory on 4-manifolds with boundary
What generalizations of Seiberg-Witten theory to 4-manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to ...
- 1,282
10
votes
2
answers
2k
views
Floer homology and Invariants for Einstein Field Equations?
Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson ...
- 17.5k
10
votes
1
answer
706
views
Moduli space of flat connections of Lie group over a 2-torus
We know that the moduli space of SU($N$) principal bundle's flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$
Namely,
$$
M_{\rm flat}=\mathbb{E} / {S}_N = \...
- 10.5k
10
votes
0
answers
1k
views
Roadmap to understanding Gromov's Non-squeezing theorem
I'm a graduate student starting out to venture into the areas of Symplectic Geometry/Topology, and was somewhat motivated by the essence of Gromov's non-squeezing theorem which in a sense made me feel ...
- 131
10
votes
0
answers
880
views
Central Yang-Mills connections, and flat connections with prescribed holonomy
Let $X$ be $\Sigma^g$ which is the Riemann surface of genus $g$, and consider a trivial $G$-bundle over it.
1) In this $2$-d setting, the space of Yang-Mills central connections is the set of ...
- 1,525
9
votes
3
answers
1k
views
Moduli spaces of connections as representation spaces
It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi_1(M), G) / G$.
Furthermore, Atiyah and Bott (1983) ...
- 5,824
9
votes
3
answers
822
views
Geometric calculations using Grassmann variables
Physicists seem to get huge computational value by introducing Grassmann variables and Grassmann integration into differential geometric calculations.
See: http://en.wikipedia.org/wiki/...
- 91
8
votes
0
answers
291
views
Infinitely many nonempty Seiberg-Witten moduli spaces
The classic "finiteness" statement in Seiberg-Witten (SW) theory is that, for any smooth closed connected 4-manifold, there are only finitely many spin-c structures with nontrivial SW ...
- 17.5k
6
votes
2
answers
740
views
Flat metrics on $n$-toruses, their systoles, and the "shortest vector problem"
Apologies if this is too basic, but I haven't been able to find any info about the question. Is there anything known about the moduli space of flat metrics on an $n$-torus (i.e., $(S^1)^n$)? ...
- 595
6
votes
0
answers
155
views
Logarithmic Darboux theorem
Let $X$ be a smooth complex analytic manifold and $D$ be a normal crossing divisor. Suppose that there is a complex analytic logarithmic symplectic structure on $X$.
Is there a Darboux like theorem ...
- 494
6
votes
0
answers
270
views
Varying a $J$-holomorphic sphere in a symplectic $4$-manifolds
I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof?
Statement. Let $(M^4,\omega)$ be a compact symlectic manifold ...
- 14.3k
5
votes
2
answers
623
views
When is differential geometry on moduli spaces possible (and productive)?
I will start out by saying I am not well-informed about moduli theory in the slightest. However, it is known that some moduli spaces (of algebro-geometric objects) have severe pathologies (Ex: Murphy'...
- 528
5
votes
1
answer
551
views
Relative logarithmic cotangent bundle
Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the ...
user149914
5
votes
1
answer
1k
views
Flat Principal Connections and Homotopy Groups?
I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...
- 1,776
4
votes
1
answer
344
views
Relative valuative criteria of properness for flat morphisms
Let $f: X\rightarrow S$ be a flat quasi-projective morphism, where $X$ is a smooth variety, and $S$ is a discrete valuation ring. Then we know that $f$ is proper morphism if and only if it satisfies ...
- 153
4
votes
1
answer
139
views
The smoothness of solutions to the Hitchin self-dual equations within a stable orbit after Sobolev completion
First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin ...
- 41
4
votes
1
answer
229
views
Orientability of moduli space and determinant bundle of ASD operator
Setting
In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections
$$\mathcal{M}_k = \{A \in L^{2}_{k}(X, \Lambda^1 \otimes\mathrm{...
- 2,533
4
votes
0
answers
109
views
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 ...
- 4,408
4
votes
0
answers
185
views
Moduli space of complex Tori [duplicate]
Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?
user21574
3
votes
1
answer
285
views
The fixed points set of the actions of $\mathbb{C}^*$ and $S^1$ on the Higgs bundle moduli space
Let $\mathcal{M}_{d}(G)$ be the moduli space of $G$-Higgs bundles. $\mathcal{M}_{d}(G)$ have a non-trivial $\mathbb{C}^{*}$-holomorphic action by multiplication of the Higgs field,
$$
z\cdot (E, \...
- 311
3
votes
1
answer
465
views
de-Rham moduli space over a compact Riemann surface
Let $X$ be a smooth projective curve over $\mathbb C$ and $M_{dR}$ denote the moduli space of stable $\Lambda$-connections of fixed rank and degree $0$. Is it known whether $M_{dR}$ is a smooth ...
- 31
3
votes
1
answer
959
views
Isomorphism classes of line bundles with connections
Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...
- 587
3
votes
1
answer
578
views
Hilbert Mumford Criterion
Let $V$ be a vector space over a field $k$. Consider the natural action of $SL(V)$ on Sym$^2 V$. Is there a easy formulation of Hilbert Mumford Criterion for semi-stablity of the points of Sym$^2 V$ ...
user100841
3
votes
0
answers
233
views
Kodaira-Spencer map in logarithmic geometry
Can anyone provide a reference for the Kodaira-Spencer map in the logarithmic geometry setting?
- 494
3
votes
0
answers
152
views
Riemannian metric over moduli space of Riemann spheres with n punctures
In the paper `Tessellations of moduli spaces and the mosaic operad' by Devadoss (https://arxiv.org/pdf/math/9807010.pdf), on page 5-6, the author identifies hyperbolic planar tree space (or the ...
- 31
3
votes
0
answers
150
views
Cubic 3-folds/genus 4 curves as an example of Kähler-Einstein moduli?
Is it currently known whether or not any the standard ball quotient models (As introduced in Allcock-Carlson-Toledo, Laza, Yokoyoma,... is an example of a moduli space of K-polystable Fano varieties (...
- 1,981
3
votes
0
answers
209
views
Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?
It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers
My question is about the moduli space of ...
user21574
2
votes
1
answer
380
views
Moduli of stable bundles - analytic approach
Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...
- 5,562
2
votes
1
answer
290
views
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
2
votes
1
answer
271
views
Canonical connection on $\mathcal{A}\times X$
Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
- 789
2
votes
2
answers
231
views
Nahm's equations with poles and conservation of characteristic polynomial
In "Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory" Nahm's equations are studied in Section 3. In particular, it is explained that their moduli space of solutions, $\...
- 1,155
2
votes
1
answer
204
views
Metric on moduli space of semistable principal G-bundles on curves
Let $X$ be an irreducible smooth projective curve over $\mathbb{C}$. Let $G$ be a connected reductive linear algebraic group over $\mathbb{C}$. Let ${\rm M}_{G,X}$ be the moduli space of semistable ...
- 523
2
votes
1
answer
1k
views
Moduli space of flat connections over a torus
Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...
- 2,816
2
votes
1
answer
612
views
pullback of poincare bundle
Let $X$ be a smooth curve and ${x,y}$ are two smooth points. Let $J_X$ be the jacobian of X i.e, the variety which parametrizes the degree 0 line bundles on X and let $\mathcal{P}$ is the poincare ...
- 151
2
votes
0
answers
170
views
A silly doubt on Log structures
Let $X=\operatorname{Spec} A$ be an affine variety. Consider the log structure given by $\mathbb N\rightarrow A$ which sends $1\mapsto 0$. Also consider the log structure $\mathbb N^r \rightarrow A$ ...
- 494
2
votes
0
answers
279
views
Symplectic form on moduli space of connections
Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it.
Is there any known construction of a bundle with a ...
- 21
1
vote
1
answer
351
views
GIT quotients of open subsets
Let X be a projective variety on with a action of reductive group G. Let L be a G Linearised ample line bundle on X. Let U be a G stable open subset of X. Let $U^{ss}:=X^{ss}\cap U$. Is it true that $...
user100841
1
vote
2
answers
912
views
Connections on the Hodge bundle?
Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...
- 1,804
1
vote
0
answers
73
views
Local Chart for Teichmuller Space as A Manifold
Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem,...
- 51
1
vote
0
answers
86
views
Banach manifold structure on the moduli space of hybrid trajectories
I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
- 791
1
vote
0
answers
114
views
On stability of coherent sheaves over a quasiprojective variety
Let $X$ be a smooth projective variety of dimension $n$ and $L$ be an ample line bundle on $X$. For any coherent sheaf $E$, one can define the first Chern class $C_1(E)$ and degree of $E$ to be $C_1(E)...
user149914
1
vote
0
answers
355
views
On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
- 494
1
vote
0
answers
501
views
The Deligne-Mumford Compactification for Closed Surfaces
I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand:
The compactified moduli space of closed ...
- 989
1
vote
0
answers
97
views
A genericity argument on family of disconnected holomorphic curves
Let $(W, \lambda)$ be an exact cobordism from $(M_+, \lambda_+)$ to $(M_-, \lambda_-)$ and $\overline{W}$ be the usual symplectic completion of $W$. Let $\mathcal{M}(J)$ be the module space of all ...
- 185