Kodaira fibres and wrapped Floer cohomology

Abstract

Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $\Omega$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,\Omega$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras; mirrors given by resolved affine surfaces at every classical bulk class for affine D, E and types II, III, IV; and mirrors given by quotient stacks of algebraic tori at root-of-unity bulk classes for the four affine stars.