TAILIEUCHUNG - Đề tài "Quantum Riemann-Roch, Lefschetz and Serre "

Given a holomorphic vector bundle E over a compact K¨hler manifold X, a 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. . | Annals of Mathematics Quantum Riemann-Roch Lefschetz and Serre By Tom Coates and Alexander Givental Annals of Mathematics 165 2007 15 53 Quantum Riemann-Roch Lefschetz and Serre By Tom Coates and Alexander Givental To Vladimir Arnold on the occassion of his 70th birthday 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 s X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H0 s f E Q H1 s 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 invariants. When E is concave and the Cx-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 duality 21 23 expresses Gromov-Witten invariants of E in terms of those of the super-manifold nE 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 c X is defined by a global section of E the genus-zero Gromov-Witten invariants of nE coincide 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 .

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