TAILIEUCHUNG - Đề tài " Hodge integrals, partition matrices, and the λg conjecture "

We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required. 0. Introduction . Overview. Let Mg,n denote the moduli space of nonsingular genus g curves with n distinct marked points (over C). | Annals of Mathematics Hodge integrals partition matrices and the Ẵg conjecture By C. Faber and R. Pandharipande Annals of Mathematics 156 2002 97 124 Hodge integrals partition matrices and the Xg conjecture By C. Faber and R. Pandharipande Abstract We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required. 0. Introduction . Overview. Let Mg n denote the moduli space of nonsingular genus g curves with n distinct marked points over C . Denote the moduli point corresponding the marked curve C p1 . pn by C p1 . . . pn c Mg n. Let WC be the canonical bundle of algebraic differentials on C. The rank g Hodge bundle E Mgn has fiber H0 C WC over C p1 . pn . The moduli space Mgn is nonsingular of dimension 3g 3 n when considered as a stack or orbifold . There is a natural compactification Mgn c Mg n by stable curves with nodal singularities . The moduli space Mg n is also a nonsingular stack. The Hodge bundle is well-defined over Mgn the fiber over a nodal curve C is defined to be the space of sections of the dualizing sheaf of C. Let Xg be the top Chern class oHỈ on Mg n. The main result of the paper is a formula for integrating tautological classes on Mg n against Xg. The study of integration against Xg has two main motivations. First such integrals arise naturally in the degree 0 sector of the Gromov-Witten theory of one-dimensional targets. The conjectural Virasoro constraints of Gromov-Witten theory predict the Xg integrals have a surprisingly simple form. Second 98 C. FABER AND R. PANDHARIPANDE the Xg integrals conjecturally govern the entire tautological ring of the moduli space M c Mg of curves of compact type. A stable curve is of compact type if the dual graph of C is a tree. . .

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