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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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