moduli spaces - Is the Torelli map an immersion?

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$H^1(T_C) to Sym^2( H^1(mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$Sym^2 ( H^0(K_C) ) to H^0( 2K_C ),$$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $gge 3$ the Torelli map OF STACKS $tau:M_gto A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $tau_{H_g}:H_gto A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

P.S. "Stacks" can be replaced everywhere by the "moduli spaces with level structure of level $lge3$ (which are fine moduli spaces).

P.P.S. The space of the first-order deformations of an abelian variety $A$ is $H^1(T_A)$. Since $T_A$ is a trivial vector bundle of rank $g$, and the cotangent space at the origin is $H^0(Omega^1_A)$, this space equals $H^1(mathcal O_A) otimes H^0(Omega^1_A)^{vee}$ and has dimension $g^2$.

A polarization is a homomorphism $lambda:Ato A^t$ from $A$ to the dual abelian variety $A^t$. It induces an isomorphism (in char 0, or for a principal polarization) from the tangent space at the origin $T_{A,0}=H^0(Omega_A^1)^{vee}$ to the tangent space at the origin $T_{A^t,0}=H^1(mathcal O_A)$. This gives an isomorphism
$$H^1(mathcal O_A) otimes H^0(Omega^1_A)^{vee} to H^1(mathcal O_A) otimes H^1(mathcal O_A).$$

The subspace of first-order deformations which preserve the polarization $lambda$ can be identified with the tensors mapping to zero in $wedge^2 H^1(mathcal O_A)$, and so is isomorphic to $Sym^2 H^1(mathcal O_A)$, outside of characteristic 2.

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