 
Summary: PROJECTIVE VARIETIES WITH BAD SEMISTABLE
REDUCTION AT 3 ONLY
VICTOR ABRASHKIN
Abstract. Suppose F = W(k)[1/p] where W(k) is the ring of
Witt vectors with coefficients in algebraicly closed field k of charac
teristic p = 2. We construct an integral theory of padic semistable
representations of the absolute Galois group of F with HodgeTate
weights from [0, p). This modification of Breuil's theory results in
the following application in the spirit of Shafarevich's Conjecture.
If Y is a projective algebraic variety over Q with good reduction
modulo all primes l = 3 and semistable reduction modulo 3 then
for the Hodge numbers of YC = Y Q C, it holds h2
(YC) = h1,1
(YC).
Introduction
Everywhere in the paper p is a fixed prime number, p = 2, k is
an algebraicly closed field of charactersitic p, F is the fraction field of
the ring of Witt vectors W(k), ¯F is a fixed algebraic closure of F and
F = Gal( ¯F/F) is the absolute Galois group of F.
Suppose Y is a projective algebraic variety over Q. Denote by YC
