 
Summary: FORMULAS
DONU ARAPURA
I want to recall some formulas which are used in the Maple/Sage scripts hodge.maple
and hodge.sage avalable in
http://www.math.purdue.edu/dvb/scripts/readme.html
Given a smooth projective variety X, the ith Betti number bi(X) = dim Hi
(X, Q)
if it's defined over C, and dim Hi
(Xet, Q ) in general. The pqth Hodge number
hpq
(X) = dim Hq
(X, p
X). These can be assembled into generating functions called
the PoincarŽe and HodgePoincarŽe polynomials
PoincareX(t) = bi(X)ti
HPoincareX(x, y) = hpq
(X)xp
yq
The Hodge decomposition implies that Poincare(t) = HPoincare(t, t) in charac
teristic 0. This may fail in positive characteristic, although it continues to hold for
