Holomorphic 1–forms on the moduli space of curves

Since the 1960s it has been well known that there are no nontrivial closed holomorphic 1–forms on the moduli space Mg of smooth projective curves of genus g > 2. We strengthen this result, proving that for g 5 there are no nontrivial holomorphic 1–forms. With this aim, we prove an extension result for sections of locally free sheaves F on a projective variety X. More precisely, we give a characterization for the surjectivity of the restriction map Formula presented for divisors D in the linear system of a sufficiently large multiple of a big and semiample line bundle L. Then we apply this to the line bundle L given by the Hodge class on the Deligne–Mumford compactification of Mg.