伴随半单轨道上的拉格朗日子流形由实形式给出

极小伴随轨道的Katzarkov-Kontsevich-Pantev猜想

Katzarkov-Kontsevich-Pantev conjecture for minimal adjoint orbits

本文证明了极小半单伴随轨道的LG模型满足关于新Hodge理论不变量的Katzarkov-Kontsevich-Pantev猜想。

Lefschetz fibrations on adjoint orbits

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.

极小伴随轨道的内蕴镜像与奇点范畴

Intrinsic mirrors for minimal adjoint orbits and categories of singularities

该研究探讨$\frak{sl}(n)$极小半单伴随轨道的Landau-Ginzburg模型镜像，验证Gross-Siebert方法的对应性，并通过构造镜像与范畴等价证明了$\frak{sl}(2)$情形下$LG(2)$的同调镜像对称。

Slodowy slices and the complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve upon this result, showing that each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits (i.e. orbits of maximal dimension). Our approach incorporates the theory of regular $\mathfrak{sl}_2$-triples and associated Slodowy slices, as developed …

超Kähler结构在超李泊利流形的叶上

Hyperkähler structures on leaves of hyper-Lie Poisson manifolds

本文探讨超Kähler结构在超李泊利流形叶上的两种不同构造方法的异同及联系。

On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits

We give a criterion for the Kostant-Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebras.

双有理等价的朗之万-金斯堡模型在余切丛和伴随轨道上

Birationally equivalent Landau-Ginzburg models on cotangent bundles and adjoint orbits

研究了最小半单伴随轨道上的李势能与庞特里亚金势能的双有理等价性，并探讨了变形家族中朗之万-金斯堡模型的变形对镜像的影响。

Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on of one the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler m…

Some Landau--Ginzburg models viewed as rational maps

[GGSM2] showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.

Homological Mirror Symmetry for the universal centralizers I: The adjoint group case

We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a Toda space), associated to any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center). This is the first and the main part of a two-part series, dealing with $G$ of adjoint type. The general case will be proved in the forthcoming second part [Jin2].

微局部层论中的拉格朗日配边函子 I

Lagrangian cobordism functor in microlocal sheaf theory I

本文利用Nadler-Shende工作，为勒让德子流形间的拉格朗日配边构造了层范畴间的函子及其右伴随，类比接触同调映射，并推导出长正合序列和高维配边的新障碍。

A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.

Abelian Ideals and the Variety of Lagrangian Subalgebras

For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by Peterson's theorem on abelian ideals.

半单李代数的Cartan子代数的奇妙紧化

Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra

本文定义了半单李代数Cartan子代数的紧化，证明其为法诺舒伯特簇，并研究了其边界除子、仿射铺砌、Weyl群作用及上同调环。