Homological and infinite-dimensional methods in algebraic geometry

点击翻译按钮获取中文摘要

Kapranov proposes to study several infinite-dimensional algebro-geometric objects such as schemes and ind-schemes with the view of applications to problems originally motivated by mathematical physics. Thus, he proposes to relate the elliptic cohomology of a complex projective variety to its derived category of coherent sheaves. While this problem is ``finite-dimensional'' in its nature, the main technical tool here is the infinite-dimensional ind-scheme of formal loops introduced by Kapranov and Vasserot. He further proposes to develop intersection theory and the theory of projective duality for ind-schemes which would, among other things, interchange inductive and projective limits. He proposes to study algebro-geometric models of the spaces of unparametrized paths by using free Lie algebras.Applications of mathematics in several areas make it necessary to study infinite-dimensional spaces. For example, the states of a vibrating medium such as a string, form an infinite-dimensional space. In string theory in physics, such picture was proposed as describing the fundamental interactions in nature.In any approach, study of infinite-dimensional objects is highly nontrivial. The traditional approach to the situation, using analysis, presents additional difficulties such as problems of convergence of series etc. Kapranov proposes a geometric study of infinite-dimensional spaces using techniques from algebra. This allows one to concentrate on essential phenomena which in many cases can be understood using the intuition from finite dimensions and to highlight the cases when the finite-dimensional intuition needs to be modified.