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**A reconstruction theorem in quantum cohomology and quantum \(K\)-theory.**
*(English)*
Zbl 1080.14065

The authors prove that \(n\)-point, genus zero descendent Gromov-Witten invariants and quantum \(K\)-invariants of a smooth projective variety \(X\) can be reconstructed from the corresponding 1-point invariants.

Descendent Gromov-Witten invariants are intersections of cotangent line classes \(\psi_i\) with pullbacks of cohomology classes \(\gamma_i\) of X, calculated on the Kontsevich moduli stack \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) Quantum \(K\)-invariants, as described by A. Givental [Mich. Math. J. 48, Spec. Vol., 295–304 (2000; Zbl 1081.14523)] and the first author [Duke Math. J. 121, No. 3, 389–424 (2004; Zbl 1051.14064)], are the \(K\)-invariants of products of cotangent line bundles and pullbacks of \(K\)-classes on \(X\), also calculated on \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) The result is that \(n\)-point invariants can be reconstructed from 1-point invariants provided that the cohomology (resp. \(K\)-theory) classes involved belong to a subring \(R\) which is generated by elements of \(\text{Pic}(X)\), that \(R\) is nondegenerate for the cohomological (resp. \(K\)-theoretic) PoincarĂ© pairing, and that the \(n\)-point invariants involving \((n-1)\) classes in \(R\) and one class in \(R^{\perp}\) vanish. The result in Gromov-Witten theory was independently proved by A. Bertram and H. P. Kley [Topology 44, No. 1, 1–24 (2005; Zbl 1083.14064)] and extends the original reconstruction result of M. Kontsevich and Yu. I. Manin for invariants without cotangent classes [Commun. Math. Phys. 164, No. 3, 525–562 (1994; Zbl 0853.14020)].

The main ingredients in the proof are a new pair of linear equivalences in the Picard group of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) between the pullbacks \(\text{ev}^{*}_i(L)\) and \(\text{ev}^{*}_j(L)\) of a line bundle \(L \rightarrow \mathbb{P}^r\) via two different markings and the cotangent line bundles \(\psi_i\) and \(\psi_j\). The authors show that these elements of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) are related via boundary divisors parametrizing maps with reducible domains and specified splitting data. These relations are of interest in their own right.

As applications, the authors reprove the famous recursion for the number of degree \(d\) rational curves in \(\mathbb{P}^2\) passing through \((3d-1)\) points and compute some quantum \(K\)-invariants of \(\mathbb{P}^1\). The results of this paper should provide a useful tool for the further study of \(n\)-point, genus zero quantum invariants and the geometry of moduli spaces.

Descendent Gromov-Witten invariants are intersections of cotangent line classes \(\psi_i\) with pullbacks of cohomology classes \(\gamma_i\) of X, calculated on the Kontsevich moduli stack \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) Quantum \(K\)-invariants, as described by A. Givental [Mich. Math. J. 48, Spec. Vol., 295–304 (2000; Zbl 1081.14523)] and the first author [Duke Math. J. 121, No. 3, 389–424 (2004; Zbl 1051.14064)], are the \(K\)-invariants of products of cotangent line bundles and pullbacks of \(K\)-classes on \(X\), also calculated on \(\overline{\mathcal{M}}_{0,n}(X, \beta).\) The result is that \(n\)-point invariants can be reconstructed from 1-point invariants provided that the cohomology (resp. \(K\)-theory) classes involved belong to a subring \(R\) which is generated by elements of \(\text{Pic}(X)\), that \(R\) is nondegenerate for the cohomological (resp. \(K\)-theoretic) PoincarĂ© pairing, and that the \(n\)-point invariants involving \((n-1)\) classes in \(R\) and one class in \(R^{\perp}\) vanish. The result in Gromov-Witten theory was independently proved by A. Bertram and H. P. Kley [Topology 44, No. 1, 1–24 (2005; Zbl 1083.14064)] and extends the original reconstruction result of M. Kontsevich and Yu. I. Manin for invariants without cotangent classes [Commun. Math. Phys. 164, No. 3, 525–562 (1994; Zbl 0853.14020)].

The main ingredients in the proof are a new pair of linear equivalences in the Picard group of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) between the pullbacks \(\text{ev}^{*}_i(L)\) and \(\text{ev}^{*}_j(L)\) of a line bundle \(L \rightarrow \mathbb{P}^r\) via two different markings and the cotangent line bundles \(\psi_i\) and \(\psi_j\). The authors show that these elements of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r, \beta)\) are related via boundary divisors parametrizing maps with reducible domains and specified splitting data. These relations are of interest in their own right.

As applications, the authors reprove the famous recursion for the number of degree \(d\) rational curves in \(\mathbb{P}^2\) passing through \((3d-1)\) points and compute some quantum \(K\)-invariants of \(\mathbb{P}^1\). The results of this paper should provide a useful tool for the further study of \(n\)-point, genus zero quantum invariants and the geometry of moduli spaces.

Reviewer: Edward Lee (Los Angeles)

### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |