Quantum Riemann - Roch, Lefschetz and Serre
| dc.creator | Coates, Tom | |
| dc.creator | Givental, Alexander | |
| dc.date | 2001-10-13 | |
| dc.date | 2001-10-19 | |
| dc.date.accessioned | 2026-07-07T04:43:49Z | |
| dc.date.available | 2026-07-07T04:43:49Z | |
| dc.description | Given a holomorphic vector bundle $E:EX X$ over a compact Kähler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: Σ\to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(Σ, f^* E) - H^1(Σ, f^*E)$. 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 $ΠE^*X$, 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 $ΠE X$ 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. | |
| dc.description | 26 pages; in this version, we correct several errors in Appendix 2 | |
| dc.identifier | https://arxiv.org/abs/math/0110142 | |
| dc.identifier | http://arxiv.org/abs/math/0110142 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62386 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14N35; 53D45 | |
| dc.title | Quantum Riemann - Roch, Lefschetz and Serre | |
| dc.type | text |