The perverse filtration for the Hitchin fibration is locally constant (Q2658468)

The \(P=W\) conjecture, which is due to \textit{M. A. A. De Cataldo} et al. [Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)], predicts that, under Simpson's twisted nonabelian Hodge correspondence, the perverse Leray filtration on the cohomology of Dolbeault moduli space matches with the weight filtration on the cohomology of Betti moduli space. This conjecture is for curve as the base. The results showed in this paper provide some evidences towards the validity of \(P=W\) conjecture for high dimensional variety as the base. More precisely, here we follow the notations in the paper, let \(S\) be a connected quasi-projective variety, let \(\mathscr{X}\to S\) be a smooth projective morphism with fibers \(\mathscr{X}_s\) are connected, and let \(G\) be a reductive group. Then we have the Hitchin projective \(S\)-morphism \(h=h(\mathscr{X}/S,G): M_D(\mathscr{X}/S,G)\to\mathcal{V}(\mathscr{X}/S,G)\), which induces the Hitchin morphisms on fibers \(h_s=h(\mathscr{X}_s,G): M_D(\mathscr{X}_s,G)\to\mathcal{V}(\mathscr{X}_s,G)\). Denote by \(\pi_D(\mathscr{X}/S,G): M_D(\mathscr{X}/S,G)\to S\) and \(\pi_B(\mathscr{X}/S,G): M_B(\mathscr{X}/S,G)\to S\) the corresponding structure morphisms respectively, then \(\pi_D(\mathscr{X}/S,G)\) is topological locally trivial over \(S\) (Fact 4.1.2), due to \(\pi_B(\mathscr{X}/S,G)\) is analytically locally trivial over \(S\), and the nonabelian Hodge correspondence. This implies, in particular, the intersection cohomology groups \(IH^\bullet(M_D(\mathscr{X}_s,G),\mathbb{Q})\) on each \(M_D(\mathscr{X}_s,G)\), and the intersection cohomology groups \(IH^\bullet(M_B(\mathscr{X}_s,G),\mathbb{Q})\) on each \(M_B(\mathscr{X}_s,G)\), give rise to local systems \(IH_D^\bullet(\mathscr{X}/S,G)\) and \(IH_B^\bullet(\mathscr{X}/S,G)\) on \(S\), respectively. For each \(s\in S\), there exists weight filtration \(W_{B,\star}(\mathscr{X}_s,G)\) for \(IH^\bullet(M_B(\mathscr{X}_s,G),\mathbb{Q})\), and perverse Leray filtration \(P_{D,\star}^{h_s}(\mathscr{X}_s,G)\) for \(IH^\bullet(M_D(\mathscr{X}_s,G),\mathbb{Q})\). These weight filtrations match-up and give rise to locally constant subsheaves \(W_{B,\star}(\mathscr{X}/S,G)\subseteq IH_B^\bullet(\mathscr{X}/S,G)\) on \(S\) (Fact 4.1.6). This does not in priori, gives the same argument for Dolbeault side. However, the authors show that, the perverse Leray filtrations also match-up and give rise to locally constant subsheaves \(P_{D,\star}^h(\mathscr{X}/S,G)\subseteq IH_D^\bullet(\mathscr{X}/S,G)\) on \(S\) (Theorem 1.1.1), hence provide some evidences towards the validity of \(P=W\).