Heat kernels, symplectic geometry, moduli spaces and finite groups
Kefeng Liu
本文应用热核导出了辛几何中的某些局部化公式，研究了黎曼曲面上平坦联络的模空间，得到了李群间某些映射的推进测度，并求解了有限群中的方程.
In this note we apply heat kernels to derive some localization formula in sympletcic geometry, to study moduli spaces of flat connections on a Riemann surface, to obtain the push-forward measures for certain maps between Lie groups and to solve equations in finite groups.
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Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.
We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa--Witten equations, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations on a closed symplectic four-manifold for the structure group $SU(2)$ or $SO(3)$ is a smooth manifold of dimension zero for a generic choice of the perturbation para…
Prequantization of the moduli space of flat connections over a four-manifold
本文在四维流形上G-主丛的联络空间上引入了一种辛结构，并给出了在边界上平凡的规范变换群在其上的Hamilton作用.辛约化成为流形上平坦联络的模空间。在平坦联络的模空间上，我们将构造一个厄米特线丛，其曲率由辛形式给出。这是陈-西蒙斯前量子线丛。基流形边界上的规范变换群以无穷小辛方式作用于平坦联络的模空间。这个作用量通过它的阿贝尔扩张被提升到前量子线丛。
Finite-dimensional construction of self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces
本文通过有限维全纯辛约简，在闭黎曼曲面上构建自对偶性及相关模空间的分层全纯辛结构。
In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, a 6-dimensional Seiberg-Witten invariant, from the moduli space of solutions to the equations, assuming that the moduli space is compact; and it has no reducible solutions. We prove that the moduli spaces are compact if the underlying manifold is a compact Kahler threefold. We th…
Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent bundle of the Teichmüller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification is proved to be compatible with these symplectic structures. Similar results are …
Symplectic structure of the moduli space of flat connections on a Riemann surface
研究亏格为$n$的$g$的Riemann曲面上平坦${\g}$ -联络的模空间上的标准辛结构.对于半单李代数${\g}$，我们得到了这种辛形式的显式有效公式，并证明了它可以表示为Poisson-Lie群$G^{*}$中敷料变换轨道上Kirillov辛形式的$n$副本和Poisson-Lie群$G$的Heisenberg二重上辛结构的$g$副本之和（对（$G,G^{*}$）对应于李代数${\g}$）。
A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then apply this framework to show that the moduli space $\mathcal{M}(P)$ of flat connections on a principal bundle $P$ over a compact manifold $M$ is a polysymplectic reduction of the space $\mathcal{A}(P)$ of all connections on $P$ by the action of the gauge group $\mathcal{G}$ with respect to a natural $Ω^2(M)/B^2(M)$-valued symplectic structure on $\mathcal{A}(P)…
Symplectic virtual localization of Gromov-Witten invariants
证明了稳定映射的模空间具有虚轨道结构。得到了虚定域公式的辛形式。
Global classification of curves on the symplectic plane
考虑了平面曲线的整体辛分类问题。首先，我们给出了一般平面曲线，即具有横向自相交的浸入在辛同构下的精确分类结果。然后，辛类的集合形成辛模空间，我们完全描述其整体拓扑项。对于一般的平面曲线，通过奇异点的局部辛模空间和一个全局拓扑项，明确地描述了辛同胚分类与复同胚分类之间的区别.引入了整体平面曲线的辛模空间和平面曲线奇点的局部辛模空间作为映射空间的等价物，并赋予它们自然的可微结构.
Let (M,ω) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,ω) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in …
Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.
Virtual moduli cycles and Gromov-Witten invariants of noncompact symplectic manifolds
本文构造并研究了非紧几何有界辛流形的Gromov-Witten不变量及其性质。提出并证明了GW-不变量的两个局部化公式。作为应用，我们得到了广义弦方程和膨胀方程及其变形的解。对于具有n维上同调群的辛流形，也得到了WDVV方程的更多解和上同调群上的量子积。为了实现这些目的，我们进一步发展的语言介绍了刘田描述虚模循环（定义刘田，小野，李田，阮和Siebert）。
On the symplectic volume of the moduli space of Spherical and Euclidean polygons
本文利用Witten公式研究多边形模空间的辛体积。我们建议使用这个体积作为一个固定的边长多边形的灵活性的措施。本文的主要结果是：在所有的球面多边形和欧氏多边形中，固定周长的正多边形是最灵活的;在所有的球面多边形中，边长为$π/2$的正多边形是最灵活的。
In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a holomorphic immersion, generalizing the known results for the moduli space of marked complex projective structures. Finally, we prove that the symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space …
From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvatu…
The analytic moduli space of holomorphic submersions
本研究构建了参数化紧凯勒底空间上稳定紧合全纯淹没的解析模空间，结合几何不变理论与最优辛联络的几何PDE研究，证明其为豪斯多夫复空间且带Weil-Petersson型凯勒度量，特例为紧凯勒流形向量丛模空间。
Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach
本文考虑环面$T^{2n}$上Ricci型辛联络的解析曲线$\nabla^t$，其中$\nabla^0$为标准联络.我们表明，由递归参数，如果$\nabla^t$是一个正式的曲线，这样的连接，然后存在一个正式的曲线的辛同构$ψ_t$，使$ψ_t\cdot\nabla^t$是一个正式的曲线平坦不变辛连接，所以$\nabla^t$是平坦的所有$t$。将这个结果应用于解析曲线的泰勒级数，意味着从$\nabla^0$开始的Ricci型辛联络的解析曲线也是平坦的。环面$(T^{2n},ω)$的辛同构群$G$作用在Ricci型辛联络空间$\E$上。作为研究模空间$\E/G$的一个初步工作，我们研究了在辛同胚形式曲线作用下联络形式曲线的模。
Moduli spaces of spacefilling branes in symplectic 4-manifolds
该研究针对辛4-流形中的空间填充膜结构，证明了全纯辛紧Kähler 4-流形上空间填充膜的模空间是光滑的并确定其维数，证明依托K3曲面与环面的局部Torelli定理。
We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperkähler metric or twistor deformations.
The symplectic geometry of polygons in hyperbolic 3-space
本文研究了三维双曲空间中定边长闭n维的模空间的辛几何。我们证明了这些模空间具有来自Poisson李理论的辛结构。我们通过沿对角线弯曲n维空间，在这些模空间上构造完全可积系统。本文的结果是前两位作者在JDG 44中关于欧氏空间中n边形连杆的结果的类似。通过变形证明了具有相同边长集的双曲三维空间和欧氏三维空间中的n边形连杆的模空间是辛同构的。
Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces
给出了实曲面上平坦联络的模空间上的Poisson结构和复曲面上全纯丛的模空间上的全纯Poisson结构的比较描述.后者的辛叶被分类的限制束某些因子。这可以被看作是固定了一个沿着与真实情况类似的“边界的复杂模拟”的连接的“完整性的复杂模拟”。
In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective…
Symplectic and Poisson structures of certain moduli spaces. II. Projective representations of cocompact discrete planar groups
设$G$是具有双不变度量的李群，不一定正定。本文证明了在早期文献中对闭曲面的基本群所作的某种构造可以推广到欧氏或非欧氏运动的任意无限保向余紧平面离散群$π$上，并得到（i）包含同态空间$\roman{Hom}(π,G)$的光滑流形$\Cal M$上的辛结构，此外，（ii）一个哈密顿$G$ -作用在$\Cal M$上，保持辛结构以及动量映射，使得约化空间等于表示空间$\roman{Rep}(π,G)$。更一般地说，这种构造也适用于某些射影表示空间。对于$G$紧的，所得到的表示空间继承了{ \it分层辛空间\/}的结构，使得分层具有有限的辛体积。例如，具有有理权的半稳定全纯抛物丛的{ \smc Mehta-Seshadri}模空间或与它们密切相关的空间通过{ \it在有限维中的辛约化以这种方式产生。
We consider moduli spaces of cyclic configurations of $N$ lines in a $2n$-dimensional symplectic vector space, such that every set of $n$ consecutive lines generates a Lagrangian subspace. We study geometric and combinatorial problems related to these moduli spaces, and prove that they are isomorphic to quotients of spaces of symmetric linear difference operators with monodromy $-1$. The symplectic cross-ratio is an invariant of two pairs of $1$-dimensional subspaces of a symplectic vector space. For $N = 2n+2$, the moduli space of Lagrangian configurations is parametrized by $n+1$ symplectic …
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over…
Moduli Spaces of Stable Polygons and Symplectic Structures on $\bar{M}_{0,n}$
本文在欧氏空间中引入了一些自然的基本多边形对象{ \it稳定多边形}，并构造了稳定多边形的新的模空间${\bfmit M}_{{\bf r}, ε}$作为复解析空间.非常出乎意料的是，这些新的模空间被证明是投射的，同构于亏格为0的Deligne-Mumford稳定曲线的模空间$\bar{\cM}_{0,n}$。此外，在稳定多边形的结构中内置了一些自然数据，这些数据导致了辛（凯勒）形式的家族（类）。在某种程度上，${\bfmit M}_{{\bf r}, ε}$可以被认为是$\bar{\cM}_{0,n}$的辛对应物和$\bar{\cM}_{0,n}$的Kapranov周商构造。所有这些为研究$\bar{\cM}_{0,n}$的Kähler拓扑提供了一个新的工具。
The moduli space ${\cal M}_n(\Si)$ of stable fiber bundles over a compact Riemann surface
研究了紧致黎曼曲面$\Si$上全纯纤维丛${\cal M}_n(\Si)$的模空间.给出了正则化行列式的一个公式和平凡丛上辛形式的一个公式。
K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations. Our main result relates this stability condition to Kähler geometry: we prove t…
The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces
设$G$是一个单复李群，$\alg{g}$是它的李代数，$K$是$G$的极大紧形式，$\alg{k}$是$K$的李代数.我们用$X\rightarrow \overline{X}$表示$\alg{g}$的反对合，它选出了紧形式$\alg{k}$。考虑具有三个洞的黎曼球面上满足附加条件$\overline{A(z)}=-A(\overline{z})$的平坦$\alg{g}$-值联络空间。我们称这个空间在规范群$\overline{g(z)}=g^{-1}(\overline{z})$作用上的商为平坦联络的PDMATHAMT模空间。我们证明了以下三个辛空间是同构的：1。平坦联络的双曲模空间。2.作为$K$的余伴随轨道的三重积的辛商得到的辛重数空间。3. Poisson-Lie重数空间等于$K$的轨道的三重积的Poisson商。
Geometric Prequantization of the Moduli Space of the Vortex equations on a Riemann surface
众所周知，黎曼曲面上涡旋方程解的模空间具有辛（实际上是凯勒）结构。我们明确地展示了这种辛结构，并进而展示了模空间上的一族辛（实际上是Kähler）结构$Ω_{Ψ_0}$，由黎曼曲面上的线丛的一部分$Ψ_0$参数化。接下来，我们证明了，对应于这些有一个家庭的预量子线丛${\mathcal P}_{Ψ_0} $的模空间的曲率是成比例的辛形式$Ω_{Ψ_0}$。
The moduli space of Higgs bundles can be constructed as a quotient of an infinite-dimensional space and hence admits an orbit type decomposition. In this paper, we show that the orbit type decomposition is a complex Whitney stratification such that each stratum is a complex symplectic submanifold and hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue to a Poisson bracket on the structure sheaf of the moduli space so that the moduli space is a stratified complex symplectic space
Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
本文填补了一般对称性下奇异辛流形约化理论的空白，研究了Marsden-Weinstein约化理论，提出了考虑微分形式奇异性的更大容许哈密顿函数集，并构建了新的奇异quasi-Hamiltonian空间。
The moduli space of complex Lagrangian submanifolds
继早期的论文中的微分几何结构的特殊拉格朗日子流形的模空间的卡-丘流形，我们遵循类似的方法，紧凑的复杂的拉格朗日子流形的（凯勒）复杂的辛流形。自然的几何结构上的模空间是一个特殊的凯勒度量，但我们提供了一个不同的观点对局部微分几何的这些，基于结构的子流形的$V\times V$（其中$V$是一个辛向量空间），这是拉格朗日关于两个常数辛形式。作为一个应用程序，我们使用这个观点如何hyperkähler度量Cecotti，费拉拉和Girardello相关联的一个特殊的Kähler结构适合到勒让德变换建设的Lindström和Ro \v cek。
Symplectic and Hyperkahler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field
本文证明了降维的Seiberg-Witten方程导出了Higgs场，并研究了由此导出的模空间。模空间所产生的一个子集的方程，证明是非空的一个紧凑的黎曼曲面的亏格g >= 1，产生一个家庭的模空间进行hyperkahler结构。对于整个方程组，相应的模空间不具有上述超卡勒结构，但具有自然的辛结构。对于环面的情况下，g=1，我们表明，全套方程有一个解决方案，不同于“涡解”。
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, García-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.D…
Star-Shaped Nakajima Quiver Varieties, Parabolic Higgs Bundle Moduli Spaces, and their Holomorphic Symplectic Structures
本文构造了从星形中岛奎弗簇到中心Levi抛物希格斯丛模空间的映射，证明其保持稳定性并同胚于具有平凡底层全纯结构的希格斯丛轨迹，进而证明该映射识别了两个空间上的自然全纯辛结构。
We show the existence of a symplectic structure on the moduli space of the Seiberg-Witten equations on $Σ\times Σ$ where $Σ$ is a compact oriented Riemann surface. To prequantize the moduli space, we construct a Quillen-type determinant line bundle on it and show its curvature is proportional to the symplectic form.
The Symplectic Geometry of Polygons in the 3-sphere
本文研究了3-球面上n维定边长闭模空间$M_r=M_r(\s^3)$的辛几何。我们证明了这些模空间具有通过SU（2）的对角共轭作用约化SU（2）中$n$共轭类的融合积（记为$C_r^n$）而得到的辛结构.这里$C_r^n$是一个拟哈密顿SU（2）空间。在$M_r$上构造了一个可积的Hamilton系统，其中Hamilton流是通过沿最大不相交对角线集合弯曲多边形而给出的.最后，我们证明了$M_r$上的辛结构与$M_r$规范理论描述的辛结构有关。本文的结果与双曲三维空间\cite{KMT}中的n维定边长模空间$M_r(\h^3)$和$\E^3$中的n维定边长模空间$M_r(\E^3)$的结果在三维球面上是相似的
Pseudoholomorphic curves and the symplectic isotopy problem
研究了伪全纯曲线的变形问题及伪全纯曲线的全模空间的相关几何性质。建立了全模空间具有鞍点性质的一个充分条件。对于嵌入伪全纯曲线的情形，给出了局部辛合痕问题的公式并求解。证明了CP ^2中任意两个同阶d \le 6的辛嵌入曲面Sigma_0，Sigma_1是辛同位素的.
Simultaneous Deformations of Symplectic Forms and Lagrangian Submanifolds
本文研究了辛流形与拉格朗日子流形的同时变形，证明其模空间与第二相对de Rham上同调群的邻域同构，并表明该空间的光滑性和有限维性。
Pseudo-holomorphic maps into folded symplectic four-manifolds
每个定向4-流形都有一个折叠辛结构，这反过来又决定了一个同伦类的相容殆复结构，这些结构在折叠超曲面（“fold”）上以受控的方式是不连续的。我们定义折叠全纯映射，即在折叠上不连续的伪全纯映射。折叠上的边值由隧道映射介导，隧道映射是穿孔到折叠超曲面上的H-全纯映射，其闭特征具有预定的渐近性。我们的主要结果是这个边值问题的线性化算子是Fredholm，从而我们得到了表现良好的局部有限维模空间.作为例子，我们的模空间的映射到折叠椭圆纤维化的特征，我们构造的例子度$d$有理映射到$S^4$。此外，我们还明确地给出了到$S^4$的1次有理映射的模空间，并证明了它具有自然紧化。目的是利用折叠辛结构将全纯映射的工具推广到所有的定向4-流形。
Group systems, groupoids, and moduli spaces of parabolic bundles
设$G$是一个李群，在它的李代数上有一个不变的非退化对称双线性型，设$π$是一个具有有限个穿刺的可定向（实）曲面$M$的基本群，设$\bold C$是$G$中的一个共轭类族，每个穿刺一个。一个有限维的建设用于获得一个辛结构的模空间上的平坦$G$ -丛紧$M$扩展到穿孔的情况。它产生了一个辛结构上的一定的光滑流形$\Cal M_{\bold C}$包含空间$\roman{Hom}(π,G)_{\bold C}$的同态映射到相应的共轭类的穿孔相应的生成元。它还产生了一个哈密顿$G$ -作用在$\Cal M_{\bold C}$上，使得约化空间等于表示的模空间$\roman{Rep}(π,G)_{\bold C}$。对于$G$紧的，每个这样的空间，通过有限维辛约化得到，是一个{ \it分层辛空间\/}。对于$G=U(n)$，我们得到了半稳定全纯抛物丛的模空间或与之密切相关的空间。
Remarks on symplectic twistor spaces
考虑了一些具有几乎复扭量结构的经典纤维丛，推导了它们在某些情况下的可积性，并研究了\textit{symplectic}扭量空间的\textit{self-holomorphic}截面.由此定义了一个$ω$相容复结构的模空间。我们回顾旗帜流形的理论，以研究西格尔域和其他域一样，这是纤维的所指的扭量空间。最后，在给定黎曼曲面共形坐标的基础上，导出了具有正则辛度量联络的黎曼曲面扭量的结构方程。然后我们与前面定义的模空间联系起来。
Applications of the Liouville symplectic form on the cotangent bundle of a loop group
本文证明了环路群余切丛上的 Liouville 辛结构能诱导 Riemann 面上带标架主 Higgs $G$-丛模栈及 $G$-连接模空间的辛结构，揭示了此前各作者构造的具有限阶标架的辛结构具有共同起源。
Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, ϕ)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $ϕ$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, ϕ)$ together with a section $θ\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ su…
Moduli spaces of flat connections on 2-manifolds, cobordism, and Witten's volume formulas
本文从Ginzburg-Guillemin-Karshon引入的Hamilton配边的观点出发，讨论了2-流形上平坦联络的模空间的Witten辛体积公式.
Comparing virtual fundamental classes: Gauge theoretical Gromov-Witten invariants for toric varieties
一般来说，小林-希钦对应建立了稳定代数几何对象的模空间与某个（广义）厄米-爱因斯坦方程的解的模空间之间的同构。我们相信，对于一个大类的模问题，这种对应关系尊重虚拟的基本类中定义的两个类别。我们证明了这一点在一个有趣的情况下，即模问题与辛因式分解问题，产生完整的环面品种。因此，我们的主要对象是与这个辛因式分解问题相关的“扭曲”线性sigma模型的模空间。我们描述这些模空间在规范理论和代数几何框架，我们确定相应的虚拟基本类。
Yang-Mills-Higgs theory for symplectic fibrations
本文的目的是研究一个方程组，它同时推广了Yang-Mills-Higgs理论中的涡旋方程和Gromov赝全纯曲线理论中的全纯性方程。我们从这两个理论的一些结果和定义扩展到一个共同的设置。我们引入一个功能推广杨米尔斯希格斯功能，其极小值与我们的方程的解决方案。我们证明了一个Hitchin-Kobayashi对应允许研究的解决方案的方程在Kaehler的情况下。给出了方程（扰动）解（的规范等价类）集的光滑流形结构（即所谓的模空间）。给出了模空间的紧化，推广了Gromov关于全纯曲线模的紧化。最后，我们利用模空间定义了（在一定条件下）紧辛流形的不变量，其中S ^1具有Hamilton几乎自由作用。这些不变量推广了Gromov-Witten不变量。这是作者的博士学位。论文它的一个章节包含在论文数学DG/9901076中。1999年4月提交论文后，笔者了解到K。Cieliebak，A. R. Gaio和D.萨拉蒙也（从不同的角度）得出了同样的方程，并制定了一个非常类似的程序（见math.SG/9909122）。
Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of g…
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