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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 \(H_2(X;\mathbb{Z})\), 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 \([(\Sigma,f)]\) 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 \(K^0(X)\to K^0(X_{g,n,d})\) along the diagram \[ \begin{tikzcd} X_{g,n+1,d}\rar["\mathrm{ev}_{n+1}"]\dar["\pi" '] &X\\ X_{g,n,d}\end{tikzcd} \] A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form \[ \mathbf{c}(\cdot)=\exp\left(\sum_{k=0}^\infty s_k \text{ch}_k(\cdot)\right), \] where \(\text{ch}_k\) are the components of the Chern character, and the \(s_k\) are arbitrary parameters. These data determine a cohomology class \(\mathbf{c}(E_{g,n,d})\) (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 \[ \mathcal{D}_{\mathbf{c},E}(t_0,t_1,\dots)=\exp\left(\sum_{g\geq 0}\hbar^{g-1}\mathcal{F}^g_{\mathbf{c},E}(t_0,t_1,\dots)\right), \] where \[ \mathcal{F}_{\mathbf{c},E}^g(t_0,t_1,\dots)=\sum_{n,d}\frac{Q^d}{n!}\int_{[X_{g,n,d}]}\mathbf{c}(E_{g,n,d}) (\sum_{k_1=0}^\infty(\text{ev}_1^*t_k)\psi_1^{k_1}) \cdots (\sum_{k_1=0}^\infty(\text{ev}_n^*t_k)\psi_n^{k_n}). \] Here \(Q^d\) is the representative of \(d\) in the semigroup ring of degrees of holomorphic curves in \(X\), \(t_0,t_1,\dots\) 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 \(K^0(X)\), 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 \(\mathcal{H}=H^*(X;\mathbb{Q})((z^{-1}))\) of Laurent polynomials in \(z^{-1}\) with coefficients in the cohomology of \(X\), endowed with the symplectic form \[ \Omega(\mathbf{f},\mathbf{g})=\frac{1}{2\pi i}\oint \left(\int_X\mathbf{f}(-z)\mathbf{g}(z)\right)\,dz. \] The subspace \(\mathcal{H}_+=H^*(X;\mathbb{Q})[z]\) is a Lagrangian subspace, and \((\mathcal{H},\Omega)\) is identified with the canonical symplectic structure on \(T^*\mathcal{H}_+\). Finally, given an infinitesimal symplectic transformation \(T\) of \(\mathcal{H}\), one can consider the differential operator \(\hat{T}\) of order \(\leq 2\) on functions on \(\mathcal{H}_+\), which is associated by quantization to the quadratic Hamiltonian \(\Omega(T\mathbf{f},\mathbf{f})/2\) on \(\mathcal{H}\). By the inclusion \(\mathcal{H}_+\hookrightarrow H^*(X;\mathbb{Q})[[z]]\), the operator \(\hat{T}\) acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable \(\mathbf{q}(z)=q_0+q_1z+q_2z^2+\cdots\) in \(H^*(X;\mathbb{Q})[[z]]\). By the dilaton shift, i.e., setting \(\mathbf{q}(z)=\mathbf{t}(z)-z\), with \(\mathbf{t}(z)=t_0+t_1z+t_2z^2+\cdots\), the operator \(\hat{T}\) acts on any function of \(t_0,t_1,\dots\), 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, \[ \mathcal{D}_{\mathbf{c},E}=\hat{\Delta}\mathcal{D}_X, \] where \(\Delta:\mathcal{H}\to \mathcal{H}\) is the linear symplectic transformation defined by the asymptotic expansion of \[ \sqrt{\mathbf{c}(E)}\prod_{m=1}^\infty \mathbf{c}(E\otimes L^{-m}) \] 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 \(\pi:X_{g,n+1,d}\to X_{g,n,d}\). If \(E=\mathbb{C}\) is the trivial line bundle, then \(E_{g,n,d}=\mathbb{C}\ominus \mathbf{E}_g^*\), 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 \[ \mathbf{c}^*(\cdot)=\exp\left(\sum_{k=0}^\infty(-1)^{k+1} s_k\mathrm{ch}_k(\cdot)\right), \] then \(\mathbf{c}^*(E^*)=1/\mathbf{c}(E)\), and one the following quantum version of Serre duality: \[ \mathcal{D}_{\mathbf{c}^*,E^*}(\mathbf{t}^*)=(\mathrm{sdet}\,\mathbf{c}(E))^{-\frac{1}{24}}\mathcal{D}_{\mathbf{c},E}(\mathbf{t}), \] where \(\mathbf{t}^*(z)=\mathbf{c}(E)\mathbf{t}(z)+(1-\mathbf{c}(E))z\).
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.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C40 Riemann-Roch theorems
14J33 Mirror symmetry (algebro-geometric aspects)
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