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The Quillen connection on L ⟶ M g , where L ∗ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves M g , g ≥ 5 , is uniquely determined by the condition that its curvature is the Weil–Petersson form on M g . The bundle of holomorphic connections on L has a unique holomorphic isomorphism with the bundle on M g given by the moduli stack of projective structures. This isomorphism takes the C ∞ section of the first bundle given by the Quillen connection on L to the C ∞ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of pointwise determining the fibres of the flat unitary subbundle of a polarized variation of Hodge structures (PVHS) of weight one. Starting from the associated Higgs field, and assuming the base has dimension 1, we construct a family of (smooth but possibly non-holomorphic) morphisms of vector bundles with the property that the intersection of their kernels at a general point is the fibre of the flat subbundle. We explore the first one of these morphisms in the case of a geometric PVHS arising from a family of smooth projective curves, showing that it acts as the cup-product with some sort of "second-order Kodaira-Spencer class" which we introduce, and check in the case of a family of smooth plane curves that this additional condition is non-trivial. [ABSTRACT FROM AUTHOR]

Let L, L' be two line bundles on I C i , let 0 -> L -> F -> L' -> 0 be a short exact sequence associated to HT ht <= dim Z (P) - 1. So M b b O SB C ' b sb and, since HT ht we conclude that M b b. But now h 0 (C ' b,M b) >= 2, h 0 (C ' v, b) = 0 and h 0 (C ' b, M b) = h 0 (C ' b, b) = 0 leads to a contradiction. [Extracted from the article]

In this paper, we study totally geodesic subvarieties Y ⊂ A g of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for g ≥ 4. We prove that if Y is generically contained in the Torelli locus, then dim Y ≤ (7 g − 2) / 3. [ABSTRACT FROM AUTHOR]

In this article, we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion U⊆K can be in the geometric case. More precisely, for any smooth projective curve C of genus g⩾2 and any r=0,...,g−1, we construct non‐isotrivial deformations of C over a quasi‐projective base such that rkK=r and rkU⩽g+12. [ABSTRACT FROM AUTHOR]

Let f : S → B be a fibration of curves and let f ∗ ω S / B = U ⊕ A be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of U . The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole U of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler). [ABSTRACT FROM AUTHOR]

Let f : S → B be a nonisotrivial fibered surface. We prove that the genus g, the rank uf of the unitary summand of the Hodge bundle f∗ωf , and the Clifford index cf satisfy the inequality uf ≤ g − cf. Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5, then the stronger bound uf ≤ g − cf − 1 holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth. [ABSTRACT FROM AUTHOR]

Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map Φ. More precisely, we describe the codifferential of the period map P associated to Φ 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. [ABSTRACT FROM AUTHOR]

We prove that Dolbeault formality, weakly Dolbeault formality and geometrically Dolbeault formality are not stable under small deformations of the complex structure. [ABSTRACT FROM AUTHOR]