P=W conjectures for character varieties with symplectic resolution
Camilla Felisetti, Mirko Mauri
Abstract
We establish P=W and PI=WI conjectures for character varieties with structural group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> which admit a symplectic resolution, i.e., for genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> and arbitrary rank, and genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> and rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> . We formulate the P=W conjecture for a resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is a specialization of the singular irreducible holomorphic symplectic variety of type O’Grady <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>6</mml:mn> </mml:math> .