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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.

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
Full Text: arXiv