The equivariant cohomology of hypertoric varieties and their real loci

超 тор品种、W-希尔伯特方案和科姆分支

Hypertoric varieties, $W$-Hilbert schemes, and Coulomb branches

本文研究了配备韦尔群辛作用的仿射超 тор品种的横贯等价希尔伯特方案，展示了科姆分支可通过此类希尔伯特方案或其哈密顿约简获得，并探讨了这些对象上的完整超卡拉西奥普利 metric。

Twistor spaces for hyperkaehler implosions

We study the geometry of the twistor space of the universal hyperkaehler implosion Q for SU(n). Using the description of Q as a hyperkaehler quiver variety, we construct a holomorphic map from the twistor space Z_Q of Q to a complex vector bundle over P^1, and an associated map of Q to the affine space R of the bundle's holomorphic sections. The map from Q to R is shown to be injective and equivariant for the action of SU(n) x T^{n-1} x SU(2). Both maps, from Q and from Z_Q, are described in detail for n=2 and n=3. We explain how the maps are built from the fundamental irreducible representati…

Real loci of based loop groups

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $σ$. This induces an involutive automorphism $τ$ of the based loop space $Ω(G)$. There exists a maximal torus $T\subset G$ such that the canonical action of $T\times S^1$ on $Ω(G)$ is compatible with $τ$ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of $Ω(G)$ and $Ω(G)^τ$ (fixed point set of $τ$) under the $T\times S^1$ moment map on $Ω(G)$ a…

关于Picard数为一的有理齐次簇上等变向量丛的完全交超凯勒四重簇

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

本文分类了Picard数为一的例外齐次簇上完全可约等变向量丛的零轨迹四重簇，通过霍奇数计算排除超凯勒情形，并确认仅存在Beauville--Donagi和Debarre--Voisin描述的两种超凯勒四重簇。

Real structures on moduli spaces of Higgs bundles

We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.

仿射GIT商中稳定轨迹的同伦群的横截性方法

Transversality Methods for Homotopy Groups of Stable Loci in Affine GIT Quotients

通过构造G-等变全纯向量丛并证明自然评价映射横截于零截面，在维数不等式下得到一般同伦避免不稳定和严格半稳定轨形的结论，并应用于控制论和统计模型中的连通性计算。