Let X/k be a smooth projective scheme over a field k of characteristic zero. For integers

i, j and d, let

S

i,j

d

:= {L ∈ Pic0

(X) : dim H

i

(X, Ω

j

X/k ⊗ L) ≥ d}.

This is a closed subset of Pic0

(X), and it was conjecturedby Beauville and Catanese and

proved by Lazersfeld, Green, and Simpson that each of its irreducible components is a

translate of a subabelian variety of Pic0

(X) by a torsion point. Pink and Roessler recently gave a new proof of this result using reduction modulo p techniques introduced by Deligne

and Illusie. I will discuss an attempt (with H. Esnault) to address some remaining issues

concerning the behaviour of

H

m

Hdg(L) := M

i+j=m

H

j

(X, Ω

i

X/k → L),

when L has finite order.