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 Ganatra and Pomerleano, arXiv:1611.06849.

中文翻译：

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仿射变种的辛同调环在拓扑极限下

我们构造了一个收敛到任何仿射变种X的辛同调环的可乘谱序列，其中第一页由与任何固定法线交叉压实\（（M，{\ mathbf {D}}）\）的地层相关的拓扑不变量构造而成的X。我们展示了一类广泛的对\（（M，{\ mathbf {D}}）\）（由于缺少相对全同构球或某些相对GW不变量而消失），其光谱序列退化，并且还可以拓扑地描述辛同调上的环结构的成对的广泛子类（具有类似特征）。示例应用包括：（a）在任何射影M中补体的辛同调环的完整拓扑描述，其同源性类跨越一个秩子空间的足够多的泛型充分除数的并集，（b）许多对数Calabi-Yau品种的零度辛同调环的完全加法和部分乘法计算，以及（c）许多证明辛同调被有限地生成为环的情况。结果中的关键技术要素是PSS形态的对数形式，该形式在我们的早期著作Ganatra和Pomerleano中进行了介绍，arXiv：1611.06849。