Studies on Pseudo-holomorphic Maps

Michigan State University, East Lansing MI

Investigators

Abstract

Abstract for DMS - 0104331 This project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to develop effective methods for computing Gromov-Witten invariants of symplectic manifolds and enumerative invariants of algebraic manifolds. This work builds on the P.I.'s recent work with E. Ionel on the `sympletic sum formula' for GW invariants. The first project involves extending the sum formula to the `modified GW invariants' recently defined by the P.I.'s student Junho Lee. This would make the formula applicable to K\"ahler surfaces with $p_g>0$ where there are important conjectures that are not currently approachable by GW methods. The second project is a symplectic approach to understanding physicists' predictions about the generating functions which count curves in Calabi-Yau 3-folds. The goal is to prove the predicted formulas by adapting some analytic methods C. Taubes developed to relate the Seiberg-Witten and Gromov invariants. The last two projects also relate to the sympletic sum formula. One seeks formulas expressing the relative Gromov-Witten invariants of a pair $(X,V)$ in terms of the usual GW invariants and the descendant classes of $X$ and $V$. The other proposes extending the sum formula to a more general type of symplectic sum which occurs in algebraic geometry when one considers projective linear systems. One of the most basic problems in mathematics is to determine the solutions of a system of polynomial equations, and an important first step toward that goal is to determine the NUMBER of solutions. There is an explicit formula for the number of simultaneous solutions of a set of n polynomials in n variables. One can then ask for the number of solutions for n polynomials in n-1 variables. In this case there is a free parameter, so the locus of solutions will be a union of curves. How many? This question has been systematically studied for 100 years, but only a few special cases were solved. Then, around 1990, it was realized that these problems can be translated into symplectic geometry, and then tackeled using the powerful machinery of mathematical gauge theory. (Gauge theory, originally part of physics, has been the focus of many very fruitful interactions between mathematicians and physicists over the past twenty years; it includes Yang-Mills and Seiberg-Witten theory, and String theory). This `Gromov-Witten invariant' approach led quickly to formulas answering some of the original enumerative problems, and there are clear indications that there are more to be discovered. This project is aimed toward further developing the symplectic gauge theory in order to produce additional general formulas and to meld these formulas into a coherent theory.