Double ramification cycles with target varieties

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 n i=1 a i = β c 1 (S). Consider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i. In other words, we require f * S (− n i=1 a i x i) to be the trivial line bundle on C. A compactification of the space of maps based upon 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 [28]. Several applications of the new formula are given. 1 Contents 0 Introduction 2 1 Curves with an rth root 13 2 GRR for the universal line bundle 27 3 Localization analysis 36 4 Applications 51 We define the set G g,n,β (X) of X-valued stable graphs as follows. A graph Γ ∈ G g,n,β (X) consists of the data

Higher genus relative Gromov–Witten theory and double ramification cycles

“…In other words, the constant term of coincides with the pushforward of the corresponding relative Gromov–Witten cycle defined by Li [15, 16]. In terms of the localization computation, we only need the polynomiality in [14] and then take the ‐coefficient of the localization contribution. This localization computation is discussed in detail in Lemma 4.8.…”

“…In [14], we already know that is a polynomial in in Chow cohomology. When , this corollary is an improved bound of the lowest degree after capping with the virtual class.…”

“…Recently, double ramification cycles with target varieties have been studied in [14]. Given a line bundle over a smooth projective variety and a topological type , the double ramification cycle with the target variety is defined using the moduli space of relative stable maps to rubber targets over .…”

“…Given a line bundle over a smooth projective variety and a topological type , the double ramification cycle with the target variety is defined using the moduli space of relative stable maps to rubber targets over . By [14], the double ramification cycle is equal to the constant term of the polynomial class , see Section 5 for the definition.…”

“…The formula for double ramification cycles in [13, 14] are obtained by relating it to certain Hurwitz–Hodge cycles which are also polynomials in and the constant terms coincide with the constant terms of . In Section 4, we generalize the identity between Hurwitz–Hodge cycles and double ramification cycles.…”