اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRationality of a generating function of Betti numbers of the unordered configuration spaces of a punctured torus
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Given a torus $E = S^{1} times S^{1}$, let $E^{times}$ be the open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th singular Betti number $h^{i}(mathrm{Conf}^{n}(E^{times}))$ of the unordered configuration space of $n$ points on $E^{times}$ can be computed as a coefficient of an explicit rational function in two variables. Our proof uses Delignes mixed Hodge structure on the singular cohomology $H^{i}(mathrm{Conf}^{n}(E^{times}))$ with complex coefficients, by considering $E$ as an elliptic curve over complex numbers. Namely, we show that the mixed Hodge structure of $H^{i}(mathrm{Conf}^{n}(E^{times}))$ is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement will imply our main result about the singular Betti numbers. We also compute all the mixed Hodge numbers $h^{p,q}(H^{i}(mathrm{Conf}^{n}(E^{times})))$ as coefficients of an explicit rational function in four variables.
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