Quantum Riemann - Roch, Lefschetz and Serre

Abstract

Given a holomorphic vector bundle $E:EX X$ over a compact K\"ahler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: \Sigma \to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(\Sigma, f^* E) - H^1(\Sigma, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map. When $E$ is concave, and the inverse $\CC^{\times}$-equivariant Euler class is chosen, the twisted theory yields GW-invariants of $EX$. The ``non-linear Serre duality principle'' expresses GW-invariants of $EX$ via those of the supermanifold $\Pi E^*X$, where the Euler class and $E^*$ replace the inverse Euler class and $E$. We derive from Theorem 1 the nonlinear Serre duality in a very general form (Corollary 2). When the bundle $E$ is convex, and a submanifold $Y\subset X$ is defined by a global section, the genus 0 GW-invariants of $\Pi E X$ coincide with those of $Y$. We prove a ``quantum Lefschetz hyperplane section principle'' (Theorem 2) expressing genus 0 GW-invariants of a complete intersection $Y$ via those of $X$. This extends earlier results of Y.-P. Lee and A. Gathmann and yields most of the known mirror formulas for toric complete intersections.