School of Mathematics Seminars and Lectures
Filtration of cohomology via symmetric semisimplicial spaces
by Dr. Oishee Banerjee (Hausdorff Centre for Mathematics, Bonn)
Tuesday, August 2, 2022
from
to
(Asia/Kolkata)
at AG-69
at AG-69
Description |
Abstract: Inspired by Deligne's use of the simplicial theory of hypercoverings in defining mixed Hodge structures, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Delta S$-hypercoverings- which also happen to appear in the avatar of modules over graded commutative monoids of the form $\mathit{Sym} M$ for some space $M$. For $\Delta S$-hypercoverings we construct a spectral sequence, somewhat like the $\check{\mathrm{C}}$ech-to-derived category spectral sequence, obtaining unified proof of old results and new-- like the computation of (in some cases, stable) singular cohomology (with $\mathbb{Q}$ coefficients) and \'etale cohomology (with $\mathbb{Q}_{\ell}$ coefficients) of the moduli space of degree $n$ maps $C\to \mathbb{P}^r$, $C$ a smooth projective curve of genus $g$, of unordered configuration spaces, of the moduli space of smooth sections of a fixed $\mathfrak{g}^r_d$ that is $m$-very ample for some $m$, some geoemetric Batyrev--Manin type conjectures over global function fields for weighted projective spaces etc. In the special case when a $\Delta S$-object $X_{\bullet}$ \emph{admits a symmetric semisimplicial filtration by $M$}, the derived indecomposables of $H^*(X_{\bullet})$ as a $H^*(\mathit{Sym} M)$-module (in the sense of Galatius--Kupers--Randal--Williams) give the cohomology of the space of \emph{$M$-indecomposables}. |