The symplectic form for the fiber bundles over a Riemann surface II

有限维构造的自对偶性及相关模空间在闭黎曼曲面上的分层全纯辛空间

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。

Shrinking the Fibers of a Submersion Splits the Riemann Tensor

This paper uses Karcher's formulation [Kar99] of the O'Neill tensors [O'N66,Gra67] to derive a concise formula for the family $Ω^ε$ of curvature forms obtained by shrinking the fibers of a submersion $π:M\to B$ of semi-Riemannian manifolds by a factor of $1-ε$. The formula clearly shows that as $ε$ approaches 1, $Ω^ε$ approaches the sum of the vertical curvature form $Ω^\mathrm{V}$ and the pullback $π^*Ω^B$ of the curvature form of $B$. The Gauss-Bonnet integrand $\mathrm{Pf}(Ω^ε)$ therefore approaches the wedge $\mathrm{Pf}(Ω^\mathrm{V})\wedgeπ^*\mathrm{Pf}(Ω^B)$. So if $π$ has compact fiber …

Symplectic structure of the moduli space of flat connections on a Riemann surface

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of $n$ copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group $G^{*}$ and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$ (the pair ($G,G^{*}$) corresponds to the Lie algebra ${\g}$).

具有全测地纤维和可积水平分布的黎曼浸没的水平拉普拉斯算子

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution

研究具有全测地纤维和可积水平分布的黎曼浸没的水平拉普拉斯算子的谱性质，证明其酉等价于基流形上某个无限秩平坦向量丛的扭曲拉普拉斯算子，并应用于缩放第一非零特征值的渐近行为。

以偶维球面空间形为纤维的丛及纤维向四分之一夹挤黎曼度量

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

本文证明：若光滑丛的每个纤维（实射影空间）带有连续变化的正曲率逐点强1/4夹挤度量，则丛的结构群可约化为标准圆度量的等距群。

Realizing the Teichmüller space as a symplectic quotient

Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichmüller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.

A symplectic proof of Verlinde factorization

We prove a multiplicity formula for Riemann-Roch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a Riemann surface.

Constraints on Lefschetz fibrations with four-dimensional fibers from Seiberg-Witten theory

We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplect…

Geometric Approach to the Weil-Petersson Symplectic Form

In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.

余切丛中的黎曼距离与辛嵌入

Riemannian distance and symplectic embeddings in cotangent bundle

本文利用标准球到余切丛邻域的辛嵌入定义类距离函数，证明其在单位圆盘余切丛情形恢复黎曼度量，并证明强 Viterbo 猜想及两球辛填充数界。

Lagrangian fibrations的截面在holomorphic symplectic manifolds上的研究

Sections of Lagrangian fibrations on holomorphic symplectic manifolds

本文研究了holomorphic symplectic manifolds上的Lagrangian fibration的截面，探讨了退化twistor变形的构造及其在复结构上的应用。

Contact fiber bundles

We define contact fiber bundles and investigate conditions for the existence of contact structures on the total space of such a bundle. The results are analogous to minimal coupling in symplectic geometry. The two applications are construction of K-contact manifolds generalizing Yamazaki's fiber join construction and a cross-section theorem for contact moment maps

Boundary behaviour of Weil-Petersson and fiber metrics for Riemann moduli spaces

The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown here to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knudsen-Deligne-Mumford compactification. This is accomplished by finding a full expansion for the hyperbolic metrics on the fibers of the universal curve as they approach the complete metrics on the nodal curves above the exceptional divisors and then using a push-forward theorem for…

Geometric Prequantization of the Moduli Space of the Vortex equations on a Riemann surface

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures $Ω_{Ψ_0}$ on the moduli space, parametrised by $Ψ_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{Ψ_0} $on the moduli space whose curvature is proportional to the symplectic forms $Ω_{Ψ_0}$.

由圆作用的奇异约化得到的黎曼-罗赫公式

A Riemann-Roch formula for singular reductions by circle actions

该研究针对允许矩映射取奇异值的可量子化哈密顿S¹-流形，推导了Guillemin-Sternberg原理下的不变黎曼-罗赫数公式，得到了仅依赖辛数据的奇异辛不变量相关结果。

Quillen-type bundle and geometric prequantization on moduli space of the Seiberg-Witten equations on product of Riemann surfaces

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.

On Kawai theorem for orbifold Riemann surfaces

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic from on $\mathrm{PSL}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric…

Singular Reduction and Quantization

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bun…

Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(ω, ω)> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $ω$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.

Torus fibrations and localization of index I

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.

Quillen bundle and Geometric Prequantization of Non-Abelian Vortices on a Riemann surface

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct Quillen line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrised by $Ψ_0$, a section of a certain bundle.

Localization of the Riemann-Roch character

We present a K-theoritic approach to the Guillemin-Sternberg conjecture, about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken and Tian-Zhang. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalisation for compact Lie group actions on manifolds that are not symplectic. Instead, these manifolds carry an invariant almost complex structure and an abstract moment map.

带拉格朗日纤维化与特殊凯勒几何的全纯辛流形中的全纯拉格朗日子簇

Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kahler geometry

该研究针对带拉格朗日纤维化的全纯辛凯勒流形，证明满足相交与投影条件的拉格朗日子簇是其底空间像上的环面纤维化且像为特殊凯勒的，解答了与Kapustin-Witten对偶相关的Hitchin问题。