49 research outputs found
Covering of elliptic curves and the kernel of the Prym map
Motivated by a conjecture of Xiao, we study families of coverings of elliptic
curves and their corresponding Prym map . More precisely, we describe the
codifferential of the period map associated to in terms of the
residue of meromorphic -forms and then we use it to give a characterization
for the coverings for which the dimension of is the least possibile.
This is useful in order to exclude the existence of non isotrivial fibrations
with maximal relative irregularity and thus also in order to give
counterexamples to the Xiao's conjecture mentioned above. The first
counterexample to the original conjecture, due to Pirola, is then analysed in
our framework.Comment: 21 pages, no figures. The seminal ideas at the base of this article
were born in the framework of the PRAGMATIC project of year 201
On the rank of the flat unitary summand of the Hodge bundle
Let be a non-isotrivial fibred surface. We prove that the
genus , the rank of the unitary summand of the Hodge bundle
and the Clifford index satisfy the inequality . Moreover, we prove that if the general fibre is a plane curve of degree
then the stronger bound holds. In particular,
this provides a strengthening of the bounds of \cite{BGN} and of \cite{FNP}.
The strongholds of our arguments are the deformation techniques developed by
the first author in \cite{Rigid} and by the third author and Pirola in
\cite{PT}, which display here naturally their power and depht.Comment: 19 pages, revised versio
Families of curves with Higgs field of arbitrarily large kernel
In this article, we consider the flat bundle (Formula presented.) and the kernel (Formula presented.) of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion (Formula presented.) can be in the geometric case. More precisely, for any smooth projective curve (Formula presented.) of genus (Formula presented.) and any (Formula presented.), we construct non-isotrivial deformations of (Formula presented.) over a quasi-projective base such that (Formula presented.) and (Formula presented.)
Totally geodesic subvarieties in the moduli space of curves
In this paper we study totally geodesic subvarieties
of the moduli space of principally polarized abelian varieties with respect to
the Siegel metric, for . We prove that if is generically contained
in the Torelli locus, then
Holomorphic 1-forms on the moduli space of curves
Since the sixties it is well known that there are no non-trivial closed
holomorphic -forms on the moduli space of smooth projective
curves of genus . In this paper, we strengthen such result proving that
for there are no non-trivial holomorphic -forms. With this aim, we
prove an extension result for sections of locally free sheaves on
a projective variety . More precisely, we give a characterization for the
surjectivity of the restriction map for divisors in the linear system of a sufficiently
large multiple of a big and semiample line bundle . Then, we apply
this to the line bundle given by the Hodge class on the Deligne
Mumford compactification of .Comment: New version with several improvement
Quillen connection and the uniformization of Riemann surfaces
The Quillen connection on , where
is the Hodge line bundle over the moduli stack of smooth
complex projective curves curves , , is uniquely
determined by the condition that its curvature is the Weil--Petersson form on
. The bundle of holomorphic connections on has a
unique holomorphic isomorphism with the bundle on given by the
moduli stack of projective structures. This isomorphism takes the
section of the first bundle given by the Quillen connection on
to the section of the second bundle given by the uniformization
theorem. Therefore, any one of these two sections determines the other
uniquely