{
"id": "1811.03609",
"version": "v1",
"published": "2018-11-08T18:53:44.000Z",
"updated": "2018-11-08T18:53:44.000Z",
"title": "Symplectic cohomology rings of affine varieties in the topological limit",
"authors": [
"Sheel Ganatra",
"Daniel Pomerleano"
],
"comment": "96 pages",
"categories": [
"math.SG",
"math.AG"
],
"abstract": "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].",
"revisions": [
{
"version": "v1",
"updated": "2018-11-08T18:53:44.000Z"
}
],
"analyses": {
"keywords": [
"affine variety",
"topological limit",
"degree zero symplectic cohomology rings",
"homology classes span",
"log calabi-yau varieties"
],
"note": {
"typesetting": "TeX",
"pages": 96,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}