  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 full-dimensional 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.
