Algebraic and Symplectic Geometry Seminar

Logarithmic and orbifold structures provide two independent ways to model curves in a variety with tangency along a normal crossings divisor. The associated systems of Gromov-Witten invariants benefit from complementary techniques; this has motivated extensive interest in comparing the two approaches.

I will report on work in which we establish a complete comparison which, crucially, incorporates negative tangency orders. Negative tangency orders appear naturally in the boundary splitting formalisms of both theories. As such, our comparison opens the way for the wholesale importation of techniques from one side to the other. Work of Sam Johnston uses our comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

Along the way, I will discuss the pathological geometry of negative tangency mapping spaces, and how this can be understood and controlled via tropical geometry. A crucial contribution of our work is the discovery of a "refined virtual class" on the logarithmic moduli space, which gives rise to a distinguished sector of the Gromov-Witten theory.

This is joint work with Luca Battistella and Dhruv Ranganathan.