Mikhail Zaidenberg
These notes are based on the lecture courses given at the Ruhr-Universit{ä}t-Bochum (03--08.02.1997) and at the Universit{é} Paul Sabatier (Toulouse, 08-12.01.1996).
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We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine surface S with trivial automorphism group, there exists a canonical smooth family of pairwise non isomorphic exotic affine 3-spheres parametrized by the closed points of S.
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals.
Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3-sphere admitts such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb{A}^{1}$-bundles over the punctured plane are classified up to $\mathbb{G}_{m}$-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact…
We give a simple general method of construction of affine varieties with infinitely many exotic models. In particular we show that for every d>1 there exists a Stein manifold of dimension d which has uncountably many different structures of affine variety.
We study real rational models of the euclidean plane $\mathbb{R}^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^2$ is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by givin…
Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.
By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on the subject, which emphasizes its analytic aspects and points out some open problems.
We explain how quantum affine algebra actions can be used to systematically construct "exotic" t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a…
We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories ${\mathcal D}_{2n}$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element of $sl_{2n}$ with two equal Jordan blocks. This representation allows us to enumerate the irreducible objects in the heart of the exotic $t$-structure on ${\mathcal D}_{2n}$ by crossingless matchings of $2n$ points on a circle. We also describe the algebra of endomorphisms of the direct sum of the irreducible objects.
Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In these lectures we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc. These results were originally obtained by B. Feigin and the author.
These are the notes from the summer school in Göttingen sponsored by NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields that took place in 2007. The aim was to give a short introduction on zeta functions over finite fields, focusing on moment zeta functions and zeta functions of affine toric hypersurfaces.
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one unsurmountable problem remained even in the case of two variables, which has been unsolved since 1939, that is the Jacobian Conjecture. These are notes for author's lectures on the geometry and topology of polynomials and the Jacobian Conjecture delivered at the l'Université de Bordeaux I and at Osaka University in 2003.
An exotic Springer correspondence for $F_4$
该研究利用特征2现象定义 $F_4$ 的奇异幂零锥，证明其轨道有限并构造相关Springer对应，还证明奇异Springer纤维可仿射铺砌，进而推导 $F_4$ 型不等参数仿射Hecke代数某些单模的几何分类。
Four Lectures on Euler Integrals
本文通过四讲内容系统介绍欧拉积分及其在粒子物理中的应用，并展示多面体几何、非常仿射簇、微分方程和计算代数等多种数学方法。
These are notes for a lecture series given at the Fields Institute Summer School in Geometric Representation Theory and Extended Affine Lie Algebras, held at the University of Ottawa in June 2009. We give an introduction to the geometric realization of crystal graphs via the quiver varieties of Lusztig and Nakajima. The emphasis is on motivating the constructions through concrete examples. The relation between the geometric construction of crystals and combinatorial realizations using Young tableaux is also discussed.
By the classical genus zero Sugawara construction one obtains from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type) representations of the Virasoro algebra. In this lecture first the classical construction is recalled. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in joint work with O.K. Sheinman. We were able to show that also in the higher genus, multi-point situa…
Park City lecture notes: around the inverse Galois problem
本讲义围绕逆伽罗瓦问题，介绍其相关研究，包括诺特问题、刚性方法及单有理簇的算术，基于2022年帕克城讲座。
Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres
本文研究了光滑动机球体的构造，证明了在无限完美域上，$\mathcal{X}_n \backslash \{\bullet\}$ 是exotic动机球体的模型。
These notes are based on a series of lectures given by the author at the Max Planck Institute for Mathematics in the Sciences in Leipzig. Addressed topics include affine and projective toric varieties, abstract normal toric varieties from fans, divisors on toric varieties and Cox's construction of a toric variety as a GIT quotient. We emphasize the role of toric varieties in solving systems of polynomial equations and provide many computational examples using the Julia package Oscar.jl.
Exotic hypercomplex structures on a torus do not exist
本文证明环面上不存在奇异超复结构，通过证明其小畑联络平坦并分类实环面上的完备平坦仿射结构，从而否定此类结构的存在。
In this article, we show that the categorical action of the shifted $q=0$ affine algebra can be used to construct (or induce) t-structures on the weight categories. The main idea is to interpret the exceptional collection constructed by Kapranov as convolution of Fourier-Mukai kernels in the categorical action via using the Borel-Weil-Bott theorem. In particular, when the categories are the bounded derived category of coherent sheaves on Grassmannians. The t-structure we obtain is precisely the exotic t-structure defined by Bezrukavnikov of the exceptional collections given by Kapranov. As an …
This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can use to "read off" a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings…
We study the exotic t-structure on the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m+n,n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, w…
These are lecture notes (by the first author) from a course (by the second author) given over two extended semesters at the University of Sydney. The first part provides an introduction to the Langlands correspondence from an arithmetical point of view. The second part gives enough background in geometric representation theory to understand Bezrukavnikov's equivalence, which is a categorification of Kazhdan and Lusztig's two realizations of the affine Hecke algebra.
These lecture notes cover a brief introduction into some of the algebro-geometric techniques used in the construction of BPS algebras. The first section introduces the derived category of coherent sheaves as a useful model of branes in toric Calabi-Yau three-folds. This model allows a rather simple derivation of quiver quantum mechanics describing low-energy dynamics of various brane systems. Vacua of such quantum mechanics can be identified with the critical equivariant cohomology of the moduli space of quiver representations. These are often counted by various crystal configurations. Using c…
Ramanujam's surface M is a contractible affine algebraic surface which is not homeomorphic to the affine plane. For any m>1 the product M^m is diffeomorphic to Euclidean space R^{4m}. We show that, for every m>0, M^m cannot be symplectically embedded into a subcritical Stein manifold. This gives the first examples of exotic symplectic structures on Euclidean space which are convex at infinity. It follows that any exhausting plurisubharmonic Morse function on M^m has at least three critical points, answering a question of Eliashberg. The heart of the argument involves showing a particular Lagra…
Algebra and geometry of link homology
本文基于2021年IHES暑期学校讲座，系统介绍Khovanov-Rozansky三重分级同调的定义、性质与最新进展，并阐述辫子簇、奇异曲线Hilbert概型、仿射Springer纤维及平面上点Hilbert概型等链环同调的代数几何模型。
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