Notions of purity and the cohomology of quiver moduli (Q2909615)

Let \(G\) be a smooth group scheme which acts on a scheme \(X\) of finite type over the finite field \(\mathbb{F}_{q},\) \(q\) a power of a prime \(p.\) The work under review is an analysis of some of the formulations of the term ``purity'' of schemes and related concepts, with particular attention to quiver representations. A finite-dimensional \(\mathbb{Q}_{l}\)-vector space \(V\) endowed with an endomorphism \(F\) is said to be strongly pure if each eigenvalue \(\alpha\in\mathbb{\bar{Q}}_{l}\) of \(F\) is of the form \(\alpha =q^{j}\) for some \(j\geq0\). On the other hand, \(\left( V,F\right) \) is called weakly pure pure if \(\alpha^{n}=q^{jn}\) for some \(n\).NEWLINENEWLINEFor \(l\neq p\) a prime one has the notion of a \(G\)-equivariant \(l\)-adic local system on \(X\). Let \(\mathcal{C}\) be a class of \(G\)-equivariant local systems on \(X\) such that (1) each local system is mixed and of weight bounded by a nonnegative integer \(w,\) (2) it is closed under extensions, and (3) it contains the local system \(\mathbb{Q}_{l}.\) It is shown that \(X\) is weakly (resp. strongly) pure with respect to \(\mathcal{C}\) if \(\left( H_{\text{et} }^{\ast}\left( \bar{X},\mathcal{\bar{L}}\right) ,F\right) \) is weakly (resp. strongly) pure for each \(\mathcal{L\in C},\) where \(F\) is the Frobenius. In the case where \(G\) acts trivially and \(\mathcal{C}\) is the class generated by the constant \(l\)-adic local system \(\mathbb{Q}_{l}\) then \(X\) is weakly (resp. strongly) pure with respect to \(\mathbb{Q}_{l}.\)NEWLINENEWLINELet \(\pi:X\rightarrow Y\) be a torsor under a linear algebraic group \(G\), and let \(\mathcal{C}_{X}\) (resp. \(\mathcal{C}_{Y}\)) be a class of \(l\)-adic local systems on \(X\) (resp. \(Y,\) where \(G\) acts trivially on \(Y\)). If \(\mathcal{C} _{X}\supseteq\pi^{\ast}\left( \mathcal{C}_{Y}\right) \) and \(X\) is weakly pure with respect to \(\mathcal{C}_{X},\) then \(Y\) is weakly pure with respect to \(\mathcal{C}_{Y}.\) If \(\mathcal{C}_{X}=\pi^{\ast}\left( \mathcal{C} _{Y}\right) \) and \(G\) is connected and \(Y\) is weakly pure with respect to \(\mathcal{C}_{Y},\) then \(X\) is weakly pure with respect to \(\mathcal{C}_{X}.\) Similar statements exist for the strongly pure cases provided that \(G\) is split.NEWLINENEWLINEThe results above are applied to GIT quotients of smooth varieties by connected reductive groups. Let \(X\) be a smooth variety which is being acted upon by a connected reductive group \(G\) and which has an ample, \(G\)-linearized line bundle \(L\) such that every semi-stable point of \(X\) with respect to \(L\) is stable, and \(X\) admits an equivariantly perfect stratification with the open stratum of the stable locus. Then if \(X\) is weakly pure with respect to the constant local system \(\mathbb{Q}_{l}\), so is \(X/\!/G\). Furthermore, if \(G\) is split and \(X\) is strongly pure with respect to the constant local system \(\mathbb{Q}_{l}\), so is \(X/\!/ G\).NEWLINENEWLINEThe work above is applied to moduli spaces of quiver representations. Let \(Q\) be a quiver, \(\mathbf{d}\) a dimension vector, and let \(X=\)rep\(\left( Q,\mathbf{d}\right) .\) Then \(X\) has a stratification defined by using semi-stability with respect to a fixed character \(\Theta\), and it is shown that this stratification is equivariantly perfect, i.e., the corresponding long exact sequences in equivariant cohomology break up into short exact sequences. As a consequence, if in addition each semi-stable point is stable, then \(H^{\ast}\left( M^{\Theta-s}\left( Q,\mathbf{d}\right) ,\mathbb{Q} _{l}\right) \) vanishes in all odd degrees, is strongly pure, and the number of \(\mathbb{F}_{q^{n}}\)-rational points of \(M^{\Theta-s}\left( Q,\mathbf{d} \right) \) is an integer polynomial function of \(q^{n}.\)