Tangential Quantum Cohomology of Arbitrary Order

A large chunk of the intersection theory of the moduli stacks of n-marked, genus 0 stable maps to a specified target variety X is encoded by the quantum cohomology ring of X. In particular, one can use the associativity of the ring structure to calculate Gromov-Witten invariants and, thus, in favorable circumstances, to calculate the number of rational curves in X passing through an appropriate number of points. In [Koc00] and [Koc04], J. Kock defined a generalization of the quantum cohomology ring, called tangency quantum cohomology. Kock’s construction encodes a larger chunk of the intersection theory of the stacks of genus 0 stable maps than does ordinary quantum cohomology. More specifically, it encodes the gravitational descendants or, to be more accurate, a modification of gravitational descendants (“enumerative descendants”) better suited to questions arising in enumerative geometry, such as the computation of characteristic numbers of families of rational curves. (An early avatar appeared in [EK99], which made a similar construction in the case of P2.) Kock’s tangency quantum cohomology, however, deals only with descendants that allow at most a single tautological class (i.e., the first Chern class of the cotangent line bundle) to be specified at each of the n marks. Thus its enumerative applications (e.g., as in [GKP02]) are to situations involving ordinary tangency only. In this paper we generalize further and thus complete the program: we define a higher-order tangential quantum cohomology ring which allows one to deal with enumerative descendants up to an arbitrary specified order d. We intend to explore the consequences of this structure for the purposes of enumerative geometry. (The sort of application we envision would extend the ideas of our earlier papers [CEK01] and [EK98].) In the present paper, however, we confine ourselves to constructing the ring and proving its associativity. In §1 we begin with a brief reminder of basic notions: moduli stacks of stable maps, tautological (psi) classes, and modified tautological classes. We state Kock’s “key formulas” for the way in which modified tautological classes restrict to boundary components of the moduli space. In §2 we present two push-pull results (Propositions 2.4 and 2.6) which are essential to the arguments in the sequel. In §3 we define Gromov-Witten and descendant classes, and present a series of technical lemmas for working with such classes. Section 4 contains our basic definition of the dth-order contact product and the main result (Theorem 4.3), which states that this contact product gives an associative ring structure to a set of formal power series with