  We construct a multiplicative spectral sequence converging to the symplectic
cohomology ring of any affine variety $X$, with first page built out of
topological invariants associated to strata of any fixed normal crossings
compactification $(M,\mathbf{D})$ of $X$. We exhibit a broad class of pairs
$(M,\mathbf{D})$ (characterized by the absence of relative holomorphic spheres
or vanishing of certain relative GW invariants) for which the spectral sequence
degenerates, and a broad subclass of pairs (similarly characterized) for which
the ring structure on symplectic cohomology can also be described
topologically. Sample applications include: (a) a complete topological
description of the symplectic cohomology ring of the complement, in any
projective $M$, of the union of sufficiently many generic ample divisors whose
homology classes span a rank one subspace, (b) complete additive and partial
multiplicative computations of degree zero symplectic cohomology rings of many
log Calabi-Yau varieties, and (c) a proof in many cases that symplectic
cohomology is finitely generated as a ring. A key technical ingredient in our
results is a logarithmic version of the PSS morphism, introduced in our earlier
work [GP1].
