On the geometric P=W conjecture (Q2135769)

Let \(C\) be a Riemann surface of genus \(g\) and let \(G\) be a complex reductive algebraic group. If the character variety \(M_B=M_B(C,G)\) admits a compactification \(\bar M_B\) such that the pair \((\bar M_B,\bar M_B\setminus M_B)\) is a projective reduced dlt pair, then the geometric P=W conjecture is: the dual boundary complex of \(\bar M_B\setminus M_B\) is homotopy equivalent to a sphere. The geometric P=W conjecture is first formulated by \textit{L. Katzarkov} et al. [Commun. Math. Phys. 336, No. 2, 853--903 (2015; Zbl 1314.32021)]. In this paper, the authors confirmed the geometric P=W conjecture if \(r=1\), \(g\geq 1\) or \(r\geq 1\), \(g=1\). Indeed, they gave two proofs of the main results adopting non-archimedean geometry and degenerations of compact hyperkähler manifolds. On other hand, they also compared the geometric P=W conjecture with the cohomological P=W conjecture established by \textit{M. A. A. De Cataldo} et al. [Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)]. And, under some assumption, they proved that the geometric P=W conjecture implies the cohomological P=W conjecture at the highest weight. Finally, they also obtained some topological invariants of dual boundary complex for arbitrary genus, which can be viewed as partial evidence for the geometric P=W conjecture for general cases.