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S. Carnahan ♦
S. Carnahan ♦
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Kodaira-Spencer Theory and moduli of curves.

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

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

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.

Kodaira-Spencer Theory and moduli of curves

I was looking at a paper 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.

Latexed maths.
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edit approved Aug 2, 2013 at 6:38
Michael Albanese
edit approved Aug 2, 2013 at 6:38
Michael Albanese
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I was looking at a paper of Farkas and the following confusing point came up.

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

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.

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.

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.

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David Zureick-Brown
David Zureick-Brown
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Kodaira-Spencer Theory and moduli of curves.

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.