Constructing the relative Fukaya category

TL;DR

This paper defines the relative Fukaya category for smooth complex projective varieties using stabilizing divisors, emphasizing its linearity over a multivariate power series ring and handling non-ample divisors.

Contribution

It introduces a new construction of the relative Fukaya category that accommodates non-ample divisors and is linear over a multivariate power series ring.

Findings

Construction works under semipositivity assumption.

Handles divisors supporting effective ample divisors without requiring ampleness.

Establishes linearity over a ring of multivariate power series with integer coefficients.

Abstract

We give a definition of Seidel's `relative Fukaya category', for a smooth complex projective variety, under a semipositivity assumption. We use the Cieliebak--Mohnke approach to transversality via stabilizing divisors. Two features of our construction are noteworthy: that we work relative to a normal crossings divisor that supports an effective ample divisor but need not have ample components; and that our relative Fukaya category is linear over a certain ring of multivariate power series with integer coefficients.