Second fundamental form and higher Gaussian maps.

In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus g ≥ 4 and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map μ_6g−6 is injective, hence all even Gaussian maps μ_2k are identically zero for all k > 3g − 3. We also give an estimate for the rank of μ_2k for g − 1 ≤ k ≤ 3g − 3.