Question

I was looking at a paper of Farkas and the following confusing point came up.

Let M_g be the moduli stack of smooth genus g curves and let pi:C \to M_g be the universal curve. Let F be \Omega^1_\pi \otimes \Omega^1_\pi, where \Omega^1_\pi is the sheaf of relative differentials of \pi. Then the pushforward \pi_* F is isomorphic \Omega^1_{M_g}.

Why is this true? Farkas says this follows from Kodaira-Spencer theory. I googled for a while and asked a few students, but couldn't figure this out.

Answer 1

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of M_g is R¹pi_*(C, T_C/Mg), which is Serre dual to pi_*F. The tangent sheaf is dual to what you wanted.