Questions Related to Curves on Complex Projective Manifolds

University Of Utah, Salt Lake City UT

Investigators

Abstract

The aim of this project is to further the understanding of the behavior of families of complex curves on a complex projective manifold X.This will be done by addressing both enumerative questions (e.g. How may curves are there with given degree and genus and incidences?) and deformation-theoretic questions (e.g. What deformations of curves lie in a generic deformation of X?). Some of the important tools for attacking such questions include the Atiyah-Bott localization theorem and Kuranishi's formal deformationtheory. In addition to the research component, the investigators will restart the successful WAGS (Western Algebraic Geometry) conferences. This research relates to counting problems. For example, given the set of solutions of a polynomial equation in several variables, how many objects of a certain type (e.g. lines) lie entirely inside the solution set? We further study the 'shape' of the collection of objects even in cases in which that collection of objects is no longer finite. This research is related to a celebrated modern theory in physics, called 'string theory,' which proposes to resolve the problem of unifying the fundamental forces of nature by slightly 'thickening' space-time in six independent directions. Our research studies the geometry of the (tiny) cross-section of the universe in those six new directions.