Logarithmic and orbifold Gromov-Witten invariants

Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been deployed to compute the invariants, most notably twisted stable maps, and recursive schemes based on the degeneration formula. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible (with some exceptions in the surface or toric case). The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan, and B.-Nabijou-Tseng-You) because of tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory - being insensitive to modifications of the boundary [Abramovich-Wise] - and further building on the work of Abramovich-Cadman-Marcus-Wise and Tseng-You, in an ongoing project with Nabijou and Ranganathan we show that genus zero tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the orbifold and logarithmic theories can be made to agree by imposing birational invariance on the former.