地点：海纳苑2幢204室

时间：9月24日，16:15 - 17:35

报告人：刘昊洋（加利福尼亚大学圣巴巴拉分校）

摘要:  

Pretalk: Algebraic and topological vector bundles: a survey

Abstract:Algebraic vector bundles carry both geometric and topological information. Understanding how these interact leads to two basic questions: which topological vector bundles admit algebraic structures, and how unique are those structures? Starting with elementary examples and the role of Chern classes in classification, we will give an informal introduction to motivic homotopy theory and explain how it provides an algebraic counterpart to the topological classification of vector bundles. We will then survey work of Asok–Fasel, Asok–Fasel–Hopkins, and Asok–Bachmann–Hopkins, emphasizing the main ideas, examples, and the scope of their comparison results. The discussion will lead to a particularly concrete question: must every algebraic vector bundle on a smooth contractible complex affine variety be trivial?

Research talk: Vector Bundles on Rational Topologically Contractible Affine Threefolds

Abstract: The generalized Serre question asks whether every algebraic vector bundle on a topologically contractible smooth affine complex variety is trivial. We give an affirmative answer for rational threefolds. More generally, for a topologically contractible smooth affine complex threefold X, we prove that CH^2(X)=0 whenever X admits a smooth projective compactification whose Chow group of 0-cycles is supported on a curve. This uncovers the link between the generalized van de Ven question, Bloch's conjecture and the generalized Serre question for threefolds. We also prove that every Koras-Russell threefold is rational and therefore has only trivial algebraic vector bundles, hence answer a question of Koras and Russell.