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# Susumu Tanabe

## The Hodge structure of the cohomology of algebraic varieties

First of all I would like to mention about the vanishing cohomology
associated to the Milnor fibre of the isolated hypersurface
singularity.
We review the results on the Milnor number (local and global) by
Kouchnirenko (Inventiones math. 32, No1 (1976), 1-32)
and then roughly sketch the formula of Hodge numbers given
by Danilov (FAA, 13, No.2(1979),32-47). I will draw pictures of
polyhedron and integer points for several examples.

The Hodge structure of the cohomology of an affine hypersurface
in an algebraic torus will be discussed. The notion of Erhart
polynomial will be introduced and used to calculate the Hodge numbers.
Both Hodge and weight filtrations allow purely combinatorial discription
of integer points located on a polyhedron.
The fundamental literature on this subject is several articles
by V.Batyrev (e.g. Duke v.69, No.2, 1993).