Hochster θ不变量与Hodge-Riemann双线性关系
Hochster's theta invariant and the Hodge-Riemann bilinear relations
摘要：
设R是孤立的超曲面奇点，M和N是N-生成的R-模.由于R是一个超曲面，M对N的扭模最终是周期为2的（即，Tor_i^R（M，N）同构于Tor_{i+2} ^R（M，N），其中i足够大）。由于R只有一个孤立奇点，所以当i足够大时，这些扭模是有限长的。Hochster将（M，N）对的θ不变量定义为长度（Tor_{2i} ^R（M，N））-长度（Tor_{2i+1} ^R（M，N）），其中i足够大。H.道已经证明，当R有偶数维且包含一个域时，对于所有对（M，N），θ不变量为零。本文证明了这一猜想的附加假设下，R是分次的，它的不相关的极大理想给出了孤立奇点。我们还对R的维数为奇数时的θ配对进行了细致的分析，并将其与光滑簇Proj（R）上的经典配对联系起来。
英文摘要：
Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated singularity, these torsion modules are of finite length for i sufficiently large. The theta invariant of the pair (M,N) is defined by Hochster to be length(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large. H. Dao has conjectured that the theta invariant is zero for all pairs (M,N) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj(R).
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