Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

Abstract

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$.

Minor corrections and improvements; 48 pages

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• The Grothendieck group of algebraic stacks
• The birational geometry of irregular Higgs bundles

Cited by in corpus (5)

• A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve
• Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
• Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve
• Twisted spectral correspondence and torus knots
• BPS cohomology for rank 2 degree 0 Higgs bundles (and more)