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String cohomology of a toroidal singularity. (English) Zbl 0949.14029
Let $$N$$ be a free abelian group of rank $$r,$$ let $$K$$ be a rational polyhedral cone inside $$N \otimes {\mathbb R}$$ satisfying the standard conditions, and let $$R = {\mathbb C}[K\cap N]$$ be the semigroup ring. Set $$Z_j = \sum_{i=1}^d \langle m_j, e_i \rangle \exp(2\pi\sqrt{-1}a_i) x_i,$$ $$j =1, \ldots, r,$$ where $$\{e_1, \ldots, e_d\}$$ is a set of lattice points of degree $$1$$ that lie in the cone $$K,$$ $$m_1, \ldots, m_r$$ is a basis of the vector space $$\text{ Hom}(N, {\mathbb Z}) \otimes {\mathbb C},$$ and the element $$x_i \in R$$ corresponds to $$e_i, i = 1, \ldots, d,$$ respectively. It is proved that $$Z_1, \ldots, Z_r$$ is a regular sequence in the ring $$R$$ as well as in its open part $$R^{\text{open}}.$$ In particular this implies the result by {}M. Hochster [Ann. Math. (2) 96, 318-337 (1972; Zbl 0233.14010)] that $$R$$ is a Cohen-Macaulay ring. In the note the author also describes an analog of the Poincaré duality and discusses how his results relate to the mirror symmetry and string cohomology.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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