# Quantum Riemann–Roch, Lefschetz and Serre | Annals of Mathematics

Abstract Given a holomorphic vector bundle E over a compact Kähler 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 H0(Σ,f∗E)⊖H1(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 invariants. 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 duality” [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 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 intersections.

KEYWORDS

SHARE & LIKE

### 数学年刊（Annals of Mathematics）

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Abstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Abstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Abstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt