Quantum Riemann-Roch, Lefschetz and Serre.

*(English)*Zbl 1189.14063Using 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.

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.

Reviewer: Domenico Fiorenza (Roma)

##### MSC:

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) |