Title: Rational connectivity and A^1-homotopy theory
Abstract:
Abstract: A¹-homotopy theory, developed by Morel and Voevodsky, is a theory
designed to import ideas from homotopy theory for use in algebraic geometry.
Perhaps the most visible success of this theory to date has been the proof of
some old conjectures of Milnor. I will explain an application of A¹-homotopy
theory to a classical (i.e., 19th century) algebro-geometric question: how can
one construct and/or distinguish varieties that are rational from those that
are merely ``close" to being rational?
First, using the (solution to the) Milnor conjectures, and then, time
permitting, using some more ``geometric" ideas from A¹-homotopy theory, I will
explain how one can construct some relatively explicit (smooth complex)
algebraic varieties that are ``very close" to being rational yet not rational.
The examples are quite elementary and concrete: they arise as appropriate
families of complex quadric hypersurfaces, and even the techniques stem from
elementary ideas, which we will explain.
(Based on joint work with F. Morel)