Double ramification cycles with target varieties
Felix Janda, Rahul Pandharipande, Aaron Pixton, Dimitri Zvonkine
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.