A Lefschetz formula for higher rank

A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.

The prime geodesic theorem for higher rank spaces

The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.

A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of C.M. abelian varieties

We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over $\bf Q$ is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in the first paper of the series. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety.

Localized Index and $L^2$-Lefschetz fixed point formula for orbifolds

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed point formula for orbifolds. These cohomological formulae for the localized indices give rise to a class of refined topological invariants…

An analytic approach to Lefschetz and Morse theory on stratified pseudomanifolds

We develop an analytic framework for Lefschetz fixed point theory and Morse theory for Hilbert complexes on stratified pseudomanifolds. We develop formulas for both global and local Lefschetz numbers and Morse, Poincaré polynomials as (polynomial) supertraces over cohomology groups of Hilbert complexes, developing techniques for relating local and global quantities using heat kernel and Witten deformation based methods. We focus on the case where the metric is wedge and the Hilbert complex is associated to a Dirac-type operator and satisfies the Witt condition, constructing Lefschetz versions …

Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

Lefschetz distribution of Lie foliations

Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural Lie group $G$. Its transverse Lie structure can be considered as a transverse action $Φ$ of $G$ on $(M,\mathcal F)$; i.e., an ``action'' which is defined up to leafwise homotopies. This $Φ$ induces an action $Φ^*$ of $G$ on the reduced leafwise cohomology $\bar H(\mathcal F)$. By using leafwise Hodge theory, the supertrace of $Φ^*$ can be defined as a distribution $L_{dis}(\mathcal F)$ on $G$ called the Lefschetz distribution of $\mathcal F$. A distributional version of the Gauss-Bonett theorem is proved, which descri…

关于Γ-恰当流形上的更高莱夫舍茨公式

Higher Lefschetz formulas on Γ-proper manifolds

本文提出了一种在Γ-恰当流形上应用热核技术，推导出狄拉克算子的更高莱夫舍茨公式的几何方法。

非紧不动点集的 Lefschetz 不动点公式

A Lefschetz fixed-point formula for noncompact fixed-point sets

本文针对完备 Riemannian 流形上的等距映射，提出了适用于非紧不动点集的等变指标定理（Lefschetz 不动点公式），通过引入局部化泛函与渐近局部算子代数，证明了已有指标为其特例并得到了正数量曲率的障碍。

局部对称空间的 Lefschetz 公式

Lefschetz formula for locally symmetric spaces

本文针对实秩一局部对称空间引入群 $C^*$-代数 $C^*_r(G,\Gamma)$，刻画 Dirac 算子的 $K$-理论指标类，并借助 Hecke 对应得到 Hecke 算子 Lefschetz 数的显式公式。

奇异叶状结构 Lefschetz 数的 Atiyah-Bott 公式

An Atiyah-Bott formula for the Lefschetz number of a singular foliation

本文将 Atiyah-Bott 框架推广到奇异叶状结构，用波前集条件替代简单不动点条件，建立了几何自同态 Lefschetz 数沿有限多个不变轨道闭包取迹的公式。

关于 $\mathbb{S}^3$-作用的硬莱夫谢茨性质

Hard Lefschetz property for $\mathbb{S}^3$-actions

本文将硬莱夫谢茨性质推广到几乎自由的 $\mathbb{S}^3$-作用，证明在满足特定上同调条件（包括 3-Sasakian 流形）时两个版本等价，并给出非 3-Sasakian 但满足 HLP 的自由作用例子。

Theta函数的模性质与刚性

Rigidity via Modular Properties of Theta Functions

本文利用Liu的方法，通过Lefschetz公式和Atiyah-Bott局部化，结合theta函数的模性质，证明了特定丛上扭曲Dirac算子的刚性。

Scaling group flow and Lefschetz trace formula for laminated spaces with $p-$adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function $\hatζ(s)$ (resp. the zeta function $ζ_Y(s)$ of a smooth projective curve $Y$ over a finite field $\mathbb{F}_q$, $q=p^f$)) one could possibly associate a foliated Riemannian laminated space $(S_{\mathbb{Q}}, \mathcal{F}, g, ϕ^t)$ (resp. $(S_Y, \mathcal{F}, g, ϕ^t)$) endowed with an action of a flow $ϕ^t$ whose primitive compact orbits should correspond to the primes of $\mathbb{Q}$ (resp. $Y$). The existence of such a foliated space and flow $ϕ^t$ is still unknown except when $Y$ is an ellipt…