Asymptotic growth of Betti numbers of ordered configuration spaces of an elliptic curve (Q2147378)

The paper under review studies the rational cohomology of the ordered configuration spaces \(\mathrm{Conf}(X,n)\) on a smooth projective variety \(X\) over \(\mathbb{C}\). It focuses on the growth of the rational Betti numbers on the case in which \(X=C\) is an elliptic curve. The main result of the paper (Theorem 3.9) establishes that the \(k\)-th Betti numbers of \(\mathrm{Conf}(C,n)\) grow as a polynomial of degree exactly \(2k-2\), improving on the known upper bound of the degree \(2k\) [\textit{T. Church}, Invent. Math. 188, No. 2, 465--504 (2012; Zbl 1244.55012)]. Moreover, using the tools developed, the cohomology groups \(H^k(\mathrm{Conf}(C,n);\mathbb{Q})\) are computed for \(k<6\) and any \(n\in \mathbb{N}\). The proofs use the Kriz model \(E(x,n)\), a differential graded algebra (DGA) that codifies the rational homotopy type of \(\mathrm{Conf}(X,n)\) for \(X\) any smooth projective variety. It is proved that, when the Euler characteristic \(\chi(X)=0\), the DGA's \(\{E(X,n)\}_{n\in\mathbb{N}}\) assemble into an \(F\)-module, i.e. a functor \(E(X)\) from the category \(F\) of finite sets with all maps to the category of DGA's, and the differential \(d:E(X)\rightarrow E(X)\) is a morphism of \(F\)-modules. This structure is used to define a filtration of \(E(X,n)\) that it is strictly compatible with the differential and that makes the computation of the cohomology of \(\mathrm{Conf}(X,n)\) more accessible.