Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913234858016768 |
|---|---|
| author | Petersen, Dan Tommasi, Orsola |
| author_facet | Petersen, Dan Tommasi, Orsola |
| contents | We construct a splitting of the cohomology of configuration spaces of points on a smooth proper variety with a multiplicative Chow--Künneth decomposition. Applied to hyperelliptic curves, this shows that the hyperelliptic Torelli group acts trivially on the rational cohomology of ordered configuration spaces of points. Moreover, if $H_{g,n}$ denotes the moduli space of $n$-pointed hyperelliptic curves, the Leray spectral sequence for the forgetful map $H_{g,n} \to H_g$ degenerates immediately, in sharp contrast to the forgetful map from $M_{g,n}$ to $M_g$. This allows for new detailed calculations of the cohomology of $M_{2,n}$ for $n \leq 5$, and the stable cohomology of $H_{g,n}$ for $n \leq 5$. We also give a detailed study of the cohomology of symplectic local systems on $M_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces Petersen, Dan Tommasi, Orsola Algebraic Geometry Algebraic Topology We construct a splitting of the cohomology of configuration spaces of points on a smooth proper variety with a multiplicative Chow--Künneth decomposition. Applied to hyperelliptic curves, this shows that the hyperelliptic Torelli group acts trivially on the rational cohomology of ordered configuration spaces of points. Moreover, if $H_{g,n}$ denotes the moduli space of $n$-pointed hyperelliptic curves, the Leray spectral sequence for the forgetful map $H_{g,n} \to H_g$ degenerates immediately, in sharp contrast to the forgetful map from $M_{g,n}$ to $M_g$. This allows for new detailed calculations of the cohomology of $M_{2,n}$ for $n \leq 5$, and the stable cohomology of $H_{g,n}$ for $n \leq 5$. We also give a detailed study of the cohomology of symplectic local systems on $M_2$. |
| title | Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2401.06455 |