范畴迹与相对 Lefschetz-Verdier 公式

Kazhdan-Laumon representations of finite Chevalley groups, character sheaves and some generalization of the Lefschetz-Verdeier trace formula

In this paper we investigate the representations of reductive groups over a finite field, introduced in 1987 by D.Kazhdan and G.Laumon. We show that generically these representations are irreducible and that their character is equal to the function obtained by traces of Frobenius morphism on the stalks of the corresponding Lusztig's character sheaf. This, together with the corresponding result by G.Lusztig, implies that generic Kazhdan-Laumon representations are isomorphic to the corresponding Deligne-Lusztig representations. In the second half of the paper we give a direct geometric construct…

Categorical proof of Holomorphic Atiyah-Bott formula

Given a symmetric monoidal $(\infty,2)$-category $\mathscr E$ we promote the trace construction to a functor. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

Local terms for transversal intersections

The goal of this note is to show that in the case of transversal intersections the "true local terms" appearing in the Lefschetz trace formula equal to the "naive local terms". To prove the result we extend the method of [Va], where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of any etale sheaf to the normal cone is monodromic and the assertion that in some cases local terms are "constant in families". As an application, we get a generalization of the Deligne-Lusztig trace formula.

范畴迹的局部项

Local terms for the categorical trace

本文为 Artin 栈引入范畴化真局部项映射，证明其可加性及与紧合前推、光滑拉回和特殊化的相容性，并据此证明 Frobenius 真局部项等于朴素局部项且朴素局部项与 !-前推相容，给出经典 Grothendieck--Lefschetz 迹公式的范畴化版本。

Categorical Donaldson-Thomas theory for local surfaces

We develope $\mathbb{C}^{\ast}$-equivariant categorical Donaldson-Thomas theory for local surfaces, i.e. the total spaces of canonical line bundles on smooth projective surfaces. We introduce $\mathbb{C}^{\ast}$-equivariant DT categories for local surfaces as Verdier quotients of derived categories of coherent sheaves on derived moduli stacks of coherent sheaves on surfaces, by subcategories of objects whose singular supports are contained in unstable loci. Via Koszul duality, our construction may be regarded as certain gluing of $\mathbb{C}^{\ast}$-equivariant derived categories of factorizat…

循环覆盖：Hodge理论与范畴Torelli定理

Cyclic covers: Hodge theory and categorical Torelli theorems

本文研究带矩形Lefschetz分解基簇上的分歧循环覆盖，定义包含$\mathrm{D}^{\mathrm{b}}(Z)$的等变子范畴$\mathcal{A}_Z$，分解Kuznetsov分量，并推出低次数素Fano三维簇的范畴Torelli定理。

Cohomological Methods in Intersection Theory

These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, and constructions of characteristic classes (as $0$-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in …

Secondary Traces

We study an invariant, the secondary trace, attached to two commuting endomorphisms of a 2-dualizable object in a symmetric monoidal higher category. We establish a secondary trace formula which encodes the natural symmetries of this invariant, identifying different realizations as an iterated trace. The proof consists of elementary Morse-theoretic arguments (with many accompanying pictures included) and may be seen as a concrete realization of the cobordism hypothesis with singularities on a marked 2-torus. From this perspective, our main result identifies the secondary trace with two alterna…

Nonlinear Traces

We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and…