Analysis of J-holomorphic Curves

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AbstractProposal: DMS-9803554Principal Investigator: Thomas ParkerThis project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to develop effective methods forcomputing Gromov-Witten invariants of symplectic manifolds andenumerative invariants of algebraic manifolds. The main thrust is acontinuing project with E. Ionel on a gluing formula for Gromovinvariants under the operation of `symplectic connect sum'. The ideais to collapse the connect sum to a singular manifold, keeping trackof the holomorphic curves along the way. Parker and Ionel havealready obtained a gluing formula in special cases; these imply thewell-known recent formula of Caporaso-Harris. A general gluingformula should be an effective new tool for computing Gromov andenumerative invariants. A second part of the project seeks to foranswer the following question. Suppose C is curve that isJ-holomorphic for some non-generic J. If one perturbs J to a nearbygeneric J', how many J'-holomorphic curves are there close to C?Several well-known problems in enumerative geometry, some solved, someunsolved, reduce to this problem. Parker and Ionel have an approachbased on the `Taubes obstruction bundle' . The last part of theproject suggests using modified Gromov invariants to obtain invariantsthat count curves which exist only for special classes of almostcomplex structures.One of the most basic problems in mathematics is to determine thesolutions of a system of polynomial equations, and an important firststep toward that goal is to determine the NUMBER of solutions. Thereis an explicit formula for the number of simultaneous solutions of aset of n polynomials in n variables. One can then ask for the numberof solutions for n polynomials in n-1 variables. In this case thereis a free parameter, so the locus of solutions will be a union ofcurves. How many? This question has been systematically studied for100 years, but only a few special cases were solved. Then, around1990, it was realized that these problems can be translated intosymplectic geometry, and then tackeled using the powerful machinery ofmathematical gauge theory. (Gauge theory, originally part of physics,has been the focus of many very fruitful interactions betweenmathematicians and physicists over the past twenty years; it includesYang-Mills and Seiberg-Witten theory, and String theory). This`Gromov invariant' approach led quickly to formulas answering some ofthe original enumerative problems, and there are clear indicationsthat there are more to be discovered. This project is aimed towardfurther developing the symplectic gauge theory in order to produceadditional general formulas, and to meld these formulas into acoherent theory.