Double ramification cycles with target varieties. Issue 4 (28th October 2020) - Janda, Felix; Pandharipande, Rahul; Pixton, Aaron; Zvonkine, Dimitri
Abstract: Let X be a nonsingular projective algebraic variety over C, and let M ¯ g, n, β ( X ) be the moduli space of stable maps f : ( C, x 1, …, x n ) → X from genus  g, n ‐pointed curves C to X of degree β . Let S be a line bundle on X . Let A = ( a 1, ⋯, a n ) be a vector of integers which satisfy ∑ i = 1 n a i = ∫ β c 1 ( S ) . Consider the following condition: the line bundle f ∗ S has a meromorphic section with zeros and poles exactly at the marked points x i with orders prescribed by the integers a i . In other words, we require f ∗ S ( − ∑ i = 1 n a i x i ) to be the trivial line bundle on C . A compactification of the space of maps based on the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ¯ g, A, β ∼ ( X, S ) . The moduli space carries a virtual fundamental class [ M ¯ g, A, β ∼ ( X, S ) ] vir ∈ A ∗ M ¯ g, A, β ∼ ( X, S ) in Gromov–Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push‐forward via the forgetful morphism of [ M ¯ g, A, β ∼ ( X, S ) ] vir to M ¯ g, n, β ( X ) . In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in (Janda, Pandharipande, Pixton and Zvonkine, Publ. Math. Inst. Hautes Études Sci . 125 (2017) 221–266). Several applications of the new formula are given.