I was looking at a [paper](http://www.google.com/url?sa=t&source=web&ct=res&cd=1&ved=0CA4QFjAA&url=http%3A%2F%2Fwww.mathematik.hu-berlin.de%2F~farkas%2Fsdg-2.pdf&ei=3KTfSoXmFo-0sgPoydHhCA&usg=AFQjCNHJWSHKXWGpaxtDsY6CxCJEDGYIBg&sig2=0Yq2ygZ4ikWs5tBgAakIeA) of Farkas and the following confusing point came up.


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


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.



Kodaira-Spencer Theory and moduli of curves
By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $\mathscr{M}\_g$ is $R^1\pi\_{\ast}(\mathscr{C}, T\_{\mathscr{C}/\mathscr{M}\_g})$, which is Serre dual to $\pi\_{\ast}\mathscr{F}$. The tangent sheaf is dual to what you wanted.


