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Quantum Riemann-Roch, Lefschetz and Serre. (English) Zbl 1189.14063

Using the formalism of quantized quadratic Hamiltonians [A. B. Givental, Mosc. Math. J. 1, No. 4, 551–568 (2001; Zbl 1008.53072)], the authors are able to prove quantum versions of three classical theorems in algebraic geometry; namely, the Riemann-Roch theorem, Serre duality, and the Lefschetz hyperplane section theorem. The key ingredient consists in introducing a notion of twisted Gromov-Witten invariants of a compact projective complex manifold X; the quantum version of the aforementioned theorems can then be seen as relations between the twisted and the nontwisted Gromov-Witten theory of X.

More precisely, let X g,n,d be the moduli space of genus g, n-pointed stable maps to X of degree d, where d is an element in H 2 (X;), and let E be a holomorphic vector bundle on X. Since a point in X g,n,d is represented by a pair (Σ,f), where Σ is a complex curve and f:ΣX a holomorphic map, one can use f to pull back E on Σ and then consider the K-theory Euler character of f * E, i.e., the virtual vector space H 0 (Σ,f * E)H 1 (Σ,f * E), as the fiber over [(Σ,f)] of a virtual vector bundle E g,n,d over X g,n,d . This intuitive construction is made completely rigorous by considering K-theory push-pull K 0 (X)K 0 (X g,n,d ) along the diagram

X g,n+1,d ev n+1 XππX g,n,d

A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form

𝐜(·)=exp k=0 s k ch k (·),

where ch k are the components of the Chern character, and the s k are arbitrary parameters. These data determine a cohomology class 𝐜(E g,n,d ) (actually, a formal family of cohomology classes parametrized by the s k ) in H * (X g,n,d ;), and one can define the total (𝐜,E)-twisted descendant potential 𝒟 𝐜,E g as

𝒟 𝐜,E (t 0 ,t 1 ,)=exp g0 g-1 𝐜,E g (t 0 ,t 1 ,),

where

𝐜,E g (t 0 ,t 1 ,)= n,d Q d n! [X g,n,d ] 𝐜(E g,n,d )( k 1 =0 (ev 1 * t k )ψ 1 k 1 )( k 1 =0 (ev n * t k )ψ n k n )·

Here Q d is the representative of d in the semigroup ring of degrees of holomorphic curves in X, t 0 ,t 1 , are rational cohomology classes on X, and ψ i is the first Chern class of the universal cotangent bundle over X g,n,d corresponding to the i-th marked point of X. For E the zero element in K 0 (X), the twisted potential 𝒟 𝐜,E g reduces to 𝒟 X , the total descendant potential of X.

At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space =H * (X;)((z -1 )) of Laurent polynomials in z -1 with coefficients in the cohomology of X, endowed with the symplectic form

Ω(𝐟,𝐠)=1 2πi X 𝐟(-z)𝐠(z)dz·

The subspace + =H * (X;)[z] is a Lagrangian subspace, and (,Ω) is identified with the canonical symplectic structure on T * + . Finally, given an infinitesimal symplectic transformation T of , one can consider the differential operator T ^ of order 2 on functions on + , which is associated by quantization to the quadratic Hamiltonian Ω(T𝐟,𝐟)/2 on . By the inclusion + H * (X;)[[z]], the operator T ^ acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable 𝐪(z)=q 0 +q 1 z+q 2 z 2 + in H * (X;)[[z]]. By the dilaton shift, i.e., setting 𝐪(z)=𝐭(z)-z, with 𝐭(z)=t 0 +t 1 z+t 2 z 2 +, the operator T ^ acts on any function of t 0 ,t 1 ,, notably on the descendant potentials.

Having introduced this formalism, the authors are able to express the relation between twisted and untwisted Gromov-Witten invariants in an extremely elegant way: up to a scalar factor,

𝒟 𝐜,E =Δ ^𝒟 X ,

where Δ: is the linear symplectic transformation defined by the asymptotic expansion of

𝐜(E) m=1 𝐜(EL -m )

under the identification of the variable z with the first Chern class of the universal line bundle L. This is the quantum Riemann-Roch theorem; it explicitly determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants. The result is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family π:X g,n+1,d X g,n,d . If E= is the trivial line bundle, then E g,n,d =𝐄 g * , where 𝐄 g is the Hodge bundle, and one recovers from quantum Riemann-Roch results of D. Mumford [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)] and C. Faber, R. Pandharipande [Invent. Math. 139, No.1, 173–199 (2000; Zbl 0960.14031)] on Hodge integrals.

If 𝐜 * is the multiplicative characteristic class

𝐜 * (·)=exp k=0 (-1) k+1 s k ch k (·),

then 𝐜 * (E * )=1/𝐜(E), and one the following quantum version of Serre duality:

𝒟 𝐜 * ,E * (𝐭 * )=( sdet 𝐜(E)) -1 24 𝒟 𝐜,E (𝐭),

where 𝐭 * (z)=𝐜(E)𝐭(z)+(1-𝐜(E))z.

Finally, if E is a convex vector bundle and a submanifold YX is defined by a global section of E, then the genus zero Gromov-Witten invariants of Y can be expresssed in terms of the invariants of X twisted by the Euler class of E. These are in turn related to the untwisted Gromov-Witten invariants of X by the quantum Riemann-Roch theorem, so the authors end up with a quantum Lefschetz hyperplane section principle, expressing genus zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results [V. V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. van Straten, Acta Math. 184, No. 1, 1–39 (2000; Zbl 1022.14014); A. Bertram, Invent. Math. 142, No. 3, 487–512 (2000; Zbl 1031.14027); A. Gathmann, Math. Ann. 325, No. 2, 393–412 (2003; Zbl 1043.14016); B. Kim, Acta Math. 183, No. 1, 71–99 (1999; Zbl 1023.14028); Y.-P. Lee, Invent. Math. 145, No. 1, 121–149 (2001; Zbl 1082.14056)], and yields most of the known mirror formulas for toric complete intersections. The idea of deriving mirror formulas by applying the Grothendieck-Riemann-Roch theorem to universal stable maps is not new: according to the authors it can be traced back at least to Kontsevich’s investigations in the early 1990s, and to Faber’s and Pandharipande’s work on Hodge integrals.


MSC:
14N35Gromov-Witten invariants, quantum cohomology, etc.
14C40Riemann-Roch theorems
14J33Mirror symmetry