algebraic varieties的欧拉示性数

algebraic varieties光滑紧化的同伦理论

Homotopy theory of smooth compactifications of algebraic varieties

本文证明了光滑代数簇的光滑紧化的同伦范畴等价于特征为零的域上的光滑簇的同伦范畴。作为一个应用，我们表明，函子发送各种各样的第p步的霍奇过滤的复上同调是表示在同伦范畴的单纯预层光滑复品种。主要动机是最近应用同伦理论的Deligne-Beilinson上同调理论。

algebraic varieties的Hodge属，I

Hodge genera of algebraic varieties, I

研究了复代数簇的交同调（Hodge theoretical genus）及其相关的特征类在复代数簇的真态射下的行为.我们得到的公式，涉及（参数化家庭）的全球不变量的复代数簇$X$这样的不变量的奇点定义在$X$上的适当代数映射。这样的公式严格限制，拓扑和分析，复杂的地图，甚至光滑簇之间的奇异性。第一和第三作者在\cite{CS1, S}中也宣布了类似的结果。

On the signed Euler characteristic property for subvarieties of abelian varieties

We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincare characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvariet…

Vector bundles on algebraic varieties

We survey some recent progress in the theory of vector bundles on algebraic varieties and related questions in algebraic K-theory.

Algebraic cobordism and flag varieties

Let $X$ be an algebraic variety over $k$ such that $\bar X=X\otimes_k\bar k$ is cellular. We study torsion elements in the Chow ring $CH^*(X)$ which corresponds to $v_iy$ in the algebraic cobordism $Ω^*(\bar X)$ where $0\not=y\in CH^*(\bar X)/p$ and $v_i$ is the generator of $BP^*$ with $|v_i|=-2(p^i-1).$ In particular, we try to compute $CH^*(X)$ from $Ω^*(\bar X)$ when $X$ are twisted complete flag varieties.