Tangency quantum cohomology

Let X be a smooth projective variety. Using modified psi classes on the stack of genus zero stable maps to X, a new associative quantum product is constructed on the cohomology space of X. When X is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in X, and specialises to the usual quantum product upon resetting the parameters corresponding to the modified psi classes. For X = P2, the product is equivalent to that of the contact cohomology of Ernström-Kennedy. Introduction Let X be a smooth projective variety over the complex numbers. The Gromov-Witten invariants of X are constructed by pulling back cohomology classes to the stack of stable maps to X and integrating over the virtual fundamental class (cf. Behrend-Manin [2], Behrend [1], Li-Tian [11]). Let Φ denote the generating function for the genus zero Gromov-Witten invariants. The quantum product is defined on the cohomology space of X by taking the third derivatives of Φ as structure constants, Ti ∗ Tj := ∑ e,f Φije g ef Tf , cf. Kontsevich-Manin [10]. The associativity of this product is equivalent to the fact that Φ satisfies the WDVV equations