AAG Seminar - Mirror symmetry for $\mathbb{Q}$-Fano 3-folds

Abstract: This is a report on the PhD thesis of my student Cristian Rodriguez.

A ${\mathbb Q}$-Fano 3-fold is a projective 3-fold $Y$ with mild (terminal) singularities, $-K_Y$ ample, and Picard rank $1$. These arise as end products of Mori's minimal model program.Thousands of families are expected, whereas there are only 17 in the smooth case. 

The mirror of a ${\mathbb Q}$-Fano 3-fold is a rigid K3 fibration $(X,D) \rightarrow ({\mathbb P}^1, \infty)$ with log Calabi--Yau total space ($K_X+D=0$) such that some power of the monodromy at infinity is maximally unipotent. We describe the mirror K3 fibration explicitly for the 95 families of $\mathbb{Q}$-Fano 3-fold hypersurfaces in weighted projective space discovered by Miles Reid. These give novel rigid rational curves on moduli spaces of polarized K3 surfaces.

Host: Matt Kerr