# Double ramification cycles with target varieties

## TL;DR

This paper derives an explicit tautological formula for the push-forward of the virtual fundamental class of a moduli space of stable maps with target variety X, generalizing Pixton's double ramification cycle formula.

## Contribution

It provides a new explicit tautological formula for the push-forward of the virtual fundamental class in the context of maps to a target variety, extending Pixton's cycle to include target data.

## Key findings

- Explicit formula in tautological classes for the push-forward of the virtual fundamental class.
- Generalization of Pixton's double ramification cycle formula to target varieties.
- Applications to calculating double ramification cycles with target X.

## Abstract

Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying 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. 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. The main result of the paper is an explicit formula (in tautological classes) for the push-forward of the virtual fundamental class of the moduli space of stable maps to rubber over X via the forgetful morphism to the moduli space of stable maps to X. In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle. Applications of the new formula, viewed as calculating double ramification cycles with target X, are given.