The algebraic de Rham theorem for toric varieties

On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida’s cohomology groups for the fan. We also prove vanishing theorems for Ishida’s cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with fulldimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author. Introduction Let ∆ be a finite fan for a free Z-module N of rank r, and denote by X := TN emb(∆) the associated r-dimensional toric variety over the field C of complex numbers. We also denote by X := X the associated complex analytic space. ∗Partly supported by the Grants-in-Aid for Co-operative Research as well as Scientific Research, the Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 14M25; Secondary 14F40, 14F32, 32S60.