On the Existence of Symplectic Resolutions of Symplectic Reductions

On symplectic resolutions and factoriality of Hamiltonian reductions

Recently, Herbig--Schwarz--Seaton have shown that $3$-large representations of a reductive group $G$ give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are $\mathbb{Q}$-factorial if and only if $G$ has finite abelianization. When $G$ is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when $G$ is semi-simple. We end with some open questions.

Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

I prove the existence of slices for an action of a reductive complex Lie group on a Kähler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

Symplectic Resolutions for Quotient Singularities

We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in T…

导出临界轨迹的平移辛约化

Shifted symplectic reduction of derived critical loci

本研究针对G不变函数的导出临界轨迹，证明其具有平移矩映射且导出辛约化对应轨道栈上诱导函数的导出临界轨迹，还给出相对版本并证明导出辛约化与导出拉格朗日相交可交换。

Existence of Symplectic Surfaces

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4.

Derived Symplectic Reduction and L-Equivariant Geometry

Motivated by the cohomological construction for the BV formalism from physics, this thesis asks how to perform the intersections and quotients appearing in the BV construction. This leads to the study of the derived symplectic reduction and the extension of these classical notions for group action and infinitesimal actions of Lie algebroids. In this context, we generalize the classical notion of moment maps for global and infinitesimal actions. We then study examples of such moment maps and their derived reductions and produce new shifted examples through a procedure of derived Lagrangian inte…

Existence of Whittaker models related to four dimensional symplectic Galois representations

We show that an irreducible cuspidal automorphic representation of the group GSp(4,A), which is not CAP and whose infinite component belongs to the discrete series, is weakly equivalent to an irreducible generic automorphic cuspidal representation, whose infinite component belongs to the same archimedean local L-packet.

Notes on GIT and symplectic reduction for bundles and varieties

These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.

不变的希尔伯特方案与经典群的辛约减的光滑化

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

本文通过不变希尔伯特方案理论，构造了经典群辛约减的光滑化，并将其与简单李代数中nilpotent轨道的闭包进行比较。

Deformation quantisation of the conic and symplectic reduction of wavefunctions

We give a short review of the algebraic procedure known as deformation quantisation, which replaces a commutative algebra with a non-commutative algebra. We use this framework to examine how the objects known as wavefunctions, as known in the quantum curve literature, arise from deformation quantisation. We give an example in terms of the planar conic $-y+x^2+2 xy + y^2$, and construct an associated wavefunction. We also give an example of the symplectic reduction of a wavefunction, following a procedure from Kontsevich and Soibelman arXiv:1701.09137.

A non-existence result on symplectic semifield spreads

We prove that there do not exist non-Desarguesian symplectic semifield spreads of PG$(5,q^2)$, $q\geq 2^{14}$ even, whose associated semifield has center containing $\mathbb{F}_q$, by proving that the only $\mathbb{F}_q$-linear set of rank 6 disjoint from the secant variety of the quadric Veronese variety of PG$(5,q^2)$ is a plane with three points of the Veronese surface of PG$(5,q^6)\setminus$PG$(5,q^2)$.

Symplectic reduction at zero angular momentum

We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $\operatorname{O}_n$ on $k$ copies of $\mathbb{C}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $\operatorname{O}_n$- and $\operatorname{SO}_n$-symplectic qu…

Cohomology of symplectic reductions of generic coadjoint orbits

Let mathcal{O}_lambda be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of mathcal{O}_lambda by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of \mathcal{O}_lambda and it makes explicit the dependence on λand a parameter in Lie(T)^*.

Gr(1, n)中的辛约化与拉格朗日子流形

Symplectic reduction and lagrangian submanifolds in Gr$(1, n)$

本文研究了Gr(1, n)中由Plucker嵌入给定Kahler形式的拉格朗日子流形，利用特定矩映射下的辛约化建立与簇M_{n-k}的对应关系，证明Gr(1, n)允许超过n种不同的光滑拉格朗日子流形拓扑类型。

弓形簇作为 $T^*(G/P)$ 的辛约化

Bow varieties as symplectic reductions of $T^*(G/P)$

本文将每个具有环面不动点的弓形簇实现为部分旗簇余切丛的幂幺群辛约化，并利用该描述计算了部分弓形簇的等变上同调，回答了先前问题，同时在附录中给出了显式广义 Mirkovic-Vybornov 同构。

辛矩阵的几乎交换概形与量子哈密顿约化

Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction

本文构造辛矩阵几乎交换概形的拉格朗日子概形并证明其为完全交，同时证明微分算子代数的量子哈密顿约化同构于C型有理Cherednik代数的球面子代数。

Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.

多边形与箭图的森理论及辛约化

Mori theory and symplectic reduction for polygons and quivers

该研究关联多边形与箭图模空间双有理几何中森理论与辛约化，分别在加法、乘法侧建立森模型与各类模空间的联系，匹配森腔分解与壁跨越，并计算小模型的代数自同构群。

On Parabolic Subgroups of Symplectic Reflection Groups

Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. An…

Quasi-Hamiltonian reduction via classical Chern-Simons theory

This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, Toën, Vaquié and Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructions used in quasi-Hamiltonian reduction. Finally, we explain how a prequantization of character stacks can be obtained purely locally.