Quantum Riemann–Roch, Lefschetz and Serre

Abstract

Given a holomorphic vector bundle E over a compact Kahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ ,f ∗ E) � H 1 (Σ ,f ∗ E). Using the formalism of quantized quadratic Hamiltonians (25), we express the descendant potential for the twisted theory in terms of that for X. This result (Theorem 1) is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invari- ants. When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. Nonlinear Serre (21), (23) expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E ∗ . We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex and a submanifold Y ⊂ X is defined by a global section of E, the genus-zero Gromov-Witten invariants of ΠE coin- cide with those of Y. We establish a quantum Lefschetz hyperplane section principle (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results (4), (9), (18), (29), (33) and yields most of the known mirror formulas for toric complete intersections.