Speaker: Sergio Console (Turin, Italy)

Title: Dolbeault cohomology and stability of abelian complex structures on
nilmanifolds

Abstract: A nilmanifold is a homogeneous space M=G/D, where G is a
real simply connected nilpotent Lie  group and D is a discrete
cocompact subgroup of G. It is complex if it is endowed with an
invariant complex structure.

Some nilmanifolds provide examples of symplectic manifolds with no
Kaehler structure. Indeed Benson and Gordon proved that a complex
nilmanifold is Kaehler if and only if it is a torus.
This is a reason for the interest in computing Dolbeault cohomology.
Since nilmanifolds are homogeneous spaces, it is natural to expect
that this can be done using invariant differential forms. However, it
is still an open problem whether this is true in general.

In this seminar  will be reviewed some classes of complex structures
for which the Dolbeault cohomology is computed by means of the complex
of forms of type (p,q) on the (complexified) Lie algebra of G. Special
attention will be given to the class of abelian complex structures for
which a very strong property related to deformations holds: they admit
a locally complete family of deformations consisting entirely of
invariant complex structures. Using Kodaira Spencer theory, one can
show that the Dolbeault cohomology of small deformations of nilmanifolds
with an abelian complex structure can be still computed by invariant
forms.

The talk is based on joint work with Anna Fino (Turin) and Yat-Sun Poon
(UC Riverside, CA, USA).