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 $\Phi$. More precisely, we describe the codifferential of the period map $P$ associated to $\Phi$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the coverings for which the dimension of $\ker(dP)$ 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 $f\colon S\to B$ be a non-isotrivial fibred surface. We prove that the genus $g$, the rank $u_f$ of the unitary summand of the Hodge bundle $f_*\omega_f$ and the Clifford index $c_f$ satisfy the inequality $u_f \leq g - c_f$. Moreover, we prove that if the general fibre is a plane curve of degree $\geq 5$ then the stronger bound $u_f \leq g - c_f-1$ 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 $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y \leq (7g -2)/3$

Holomorphic 1-forms on the moduli space of curves

Since the sixties it is well known that there are no non-trivial closed holomorphic $1$-forms on the moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g>2$. In this paper, we strengthen such result proving that for $g\geq 5$ there are no non-trivial holomorphic $1$-forms. With this aim, we prove an extension result for sections of locally free sheaves $\mathcal{F}$ on a projective variety $X$. More precisely, we give a characterization for the surjectivity of the restriction map $\rho_D:H^0(\mathcal{F})\to H^0(\mathcal{F}|_{D})$ for divisors $D$ in the linear system of a sufficiently large multiple of a big and semiample line bundle $\mathcal{L}$. Then, we apply this to the line bundle $\mathcal{L}$ given by the Hodge class on the Deligne Mumford compactification of $\mathcal{M}_g$.Comment: New version with several improvement

Quillen connection and the uniformization of Riemann surfaces

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${\mathcal M}_g$. The bundle of holomorphic connections on ${\mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${\mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^\infty$ section of the first bundle given by the Quillen connection on ${\mathcal L}$ to the $C^\infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely