*(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 ${H}_{2}(X;\mathbb{Z})$, and let $E$ be a holomorphic vector bundle on $X$. Since a point in ${X}_{g,n,d}$ is represented by a pair $({\Sigma},f)$, where ${\Sigma}$ is a complex curve and $f:{\Sigma}\to X$ a holomorphic map, one can use $f$ to pull back $E$ on ${\Sigma}$ and then consider the $K$-theory Euler character of ${f}^{*}E$, i.e., the virtual vector space ${H}^{0}({\Sigma},{f}^{*}E)\ominus {H}^{1}({\Sigma},{f}^{*}E)$, as the fiber over $\left[\right({\Sigma},f\left)\right]$ 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 ${K}^{0}\left(X\right)\to {K}^{0}\left({X}_{g,n,d}\right)$ along the diagram

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

where ${\text{ch}}_{k}$ are the components of the Chern character, and the ${s}_{k}$ are arbitrary parameters. These data determine a cohomology class $\mathbf{c}\left({E}_{g,n,d}\right)$ (actually, a formal family of cohomology classes parametrized by the ${s}_{k}$) in ${H}^{*}({X}_{g,n,d};\mathbb{Q})$, and one can define the total $(\mathbf{c},E)$-twisted descendant potential ${\mathcal{D}}_{\mathbf{c},E}^{g}$ as

where

Here ${Q}^{d}$ is the representative of $d$ in the semigroup ring of degrees of holomorphic curves in $X$, ${t}_{0},{t}_{1},\cdots $ are rational cohomology classes on $X$, and ${\psi}_{i}$ 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 ${K}^{0}\left(X\right)$, the twisted potential ${\mathcal{D}}_{\mathbf{c},E}^{g}$ reduces to ${\mathcal{D}}_{X}$, the total descendant potential of $X$.

At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space $\mathscr{H}={H}^{*}(X;\mathbb{Q})\left(\left({z}^{-1}\right)\right)$ of Laurent polynomials in ${z}^{-1}$ with coefficients in the cohomology of $X$, endowed with the symplectic form

The subspace ${\mathscr{H}}_{+}={H}^{*}(X;\mathbb{Q})\left[z\right]$ is a Lagrangian subspace, and $(\mathscr{H},{\Omega})$ is identified with the canonical symplectic structure on ${T}^{*}{\mathscr{H}}_{+}$. Finally, given an infinitesimal symplectic transformation $T$ of $\mathscr{H}$, one can consider the differential operator $\widehat{T}$ of order $\le 2$ on functions on ${\mathscr{H}}_{+}$, which is associated by quantization to the quadratic Hamiltonian ${\Omega}(T\mathbf{f},\mathbf{f})/2$ on $\mathscr{H}$. By the inclusion ${\mathscr{H}}_{+}\hookrightarrow {H}^{*}(X;\mathbb{Q})\left[\left[z\right]\right]$, the operator $\widehat{T}$ acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable $\mathbf{q}\left(z\right)={q}_{0}+{q}_{1}z+{q}_{2}{z}^{2}+\cdots $ in ${H}^{*}(X;\mathbb{Q})\left[\left[z\right]\right]$. By the dilaton shift, i.e., setting $\mathbf{q}\left(z\right)=\mathbf{t}\left(z\right)-z$, with $\mathbf{t}\left(z\right)={t}_{0}+{t}_{1}z+{t}_{2}{z}^{2}+\cdots $, the operator $\widehat{T}$ acts on any function of ${t}_{0},{t}_{1},\cdots $, 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,

where ${\Delta}:\mathscr{H}\to \mathscr{H}$ is the linear symplectic transformation defined by the asymptotic expansion of

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 $\pi :{X}_{g,n+1,d}\to {X}_{g,n,d}$. If $E=\u2102$ is the trivial line bundle, then ${E}_{g,n,d}=\u2102\ominus {\mathbf{E}}_{g}^{*}$, where ${\mathbf{E}}_{g}$ 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 ${\mathbf{c}}^{*}$ is the multiplicative characteristic class

then ${\mathbf{c}}^{*}\left({E}^{*}\right)=1/\mathbf{c}\left(E\right)$, and one the following quantum version of Serre duality:

where ${\mathbf{t}}^{*}\left(z\right)=\mathbf{c}\left(E\right)\mathbf{t}\left(z\right)+(1-\mathbf{c}\left(E\right))z$.

Finally, if $E$ is a convex vector bundle and a submanifold $Y\subset X$ 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:

14N35 | Gromov-Witten invariants, quantum cohomology, etc. |

14C40 | Riemann-Roch theorems |

14J33 | Mirror symmetry |