Cell decomposition and dual boundary complexes of character varieties

2023/07/31 by Su, Tao
#14F45 (Secondary) #14M35 (Primary) 14C30 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2307.16657

Abstract

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson asserts that for any smooth Betti moduli space M_B of complex dimension d over a punctured Riemann surface, the dual boundary complex \mathbbD\partialM_B is homotopy equivalent to a (d-1)-dimensional sphere. Here, we consider M_B as a generic GL_n(ℂ)-character variety defined on a Riemann surface of genus g, with local monodromies specified by generic semisimple conjugacy classes at k punctures. In this article, we establish the weak geometric P=W conjecture for all very generic M_B in the sense that at least one conjugacy class is regular semisimple. A crucial step is to establish a stronger form of A. Mellit's cell decomposition theorem, i.e. we decompose M_B (without passing to a vector bundle) into locally closed subvarieties of the form (ℂ^×)^d-2b\timesA, where A is stably isomorphic to ℂ^b. A second ingredient involves a motivic characterization of the integral cohomology of dual boundary complexes developed in a subsequent article [Su24]. Following C. Simpson's strategy, the proof is now an inductive computation of the dual boundary complexes from such a cell decomposition.

Related

• Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity
• Cell decompositions of character varieties
• Integral cohomology of dual boundary complexes is motivic
• On the geometric P=W conjecture
• The dual boundary complex of the SL₂ character variety of a punctured sphere