Constancy of Jacobi osculating rank of g.o. spaces of compact and non-compact type

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

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Normal functions and the height of Gross-Schoen cycles

Nagoya Mathematical Journal ◽

10.1215/00277630-2413391 ◽

2014 ◽

Vol 214 ◽

pp. 53-77 ◽

Cited By ~ 1

Author(s):

Robin De Jong

Keyword(s):

Line Bundle ◽

Moduli Space ◽

The Self ◽

Compact Type ◽

Stable Curves ◽

Normal Functions ◽

Chern Form ◽

Bloch Line ◽

Pointed Curve ◽

Dualizing Sheaf

AbstractWe prove a variant of a formula due to Zhang relating the Beilinson– Bloch height of the Gross–Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach, the height of the Gross–Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is nonnegative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal Jacobian.

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Realizations of Rank One Symmetric Spaces of Compact Type

Springer Monographs in Mathematics - Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group ◽

10.1007/978-1-84882-533-8_3 ◽

2009 ◽

pp. 61-83

Author(s):

Valery V. Volchkov ◽

Vitaly V. Volchkov

Keyword(s):

Symmetric Spaces ◽

Compact Type ◽

Rank One

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