Constructing the relative Fukaya category

Abstract

We give a definition of Seidel’s ‘relative Fukaya category’, for a smooth complex projective variety relative to a simple normal crossings divisor, 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 which 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.