Abstract

Consider the log K3 surface given by the pair of the projective plane and an elliptic curve. Based on rather intricate computations, N. Takahashi stated 3 influencial conjectures in 2001. They are (1) the log-local correspondence, (2) the BPS log-local correspondence and (3) the B-model to the log K3 surface. While (1) was proven for the plane shortly afterwards by Gathmann, a proof in all generality only came in 2017 with subsequent ongoing variations on the theme. The proof of (2) was completed in 2019 through a sequence of works by Gräfnitz and Bousseau that necessitated the study of wall-crossing in spaces of Bridgeland stability conditions, a non-existent notion in 2001. Generalisations to other log K3 surfaces are currently out of reach. My talk will be about (3), which we prove in joint work with Ruddat and Siebert. By using the mirror construction of the Gross-Siebert programme, we obtain exactly the prediction by Takahashi from more than 20 years ago.