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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]

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]