论文标题
$ \ bar {\ mathcal {m}} _ {g，n} $
The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
论文作者
论文摘要
我们证明，有理共同体$ h^{11}（\ bar {\ mathcal {m}}} _ {g，n}）$消失，除非$ g = 1 $和$ n \ geq 11 $。我们还表明，对于所有$ h^k（\ bar {\ mathcal {m}} _ {g，n}）$对于所有$ k \ leq 12 $来说都是纯粹的hodge-tate，并将$ \＃\ bar {\ mathcal {\ mathcal {m}}} _ {g，niry y mathies \ imaties \ imaties \ imaties \ bar}（ $ q $中的多项式。 In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.
We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.
