Intersection cohomology of moduli spaces of vector bundles over curves

Abstract

We compute the intersection cohomology of the moduli spaces $M_{r,d}$ of semistable vector bundles having rank $r$ and degree $d$ over a curve. We do this by relating the Hodge-Deligne polynomial of the intersection cohomology of $M_{r,d}$ to the Donaldson-Thomas invariants of the curve. These invariants can be computed by methods going back to Harder, Narasimhan, Desale and Ramanan. More generally, we introduce Donaldson-Thomas classes in the Grothendieck group of mixed Hodge modules over $M_{r,d}$ and relate them to the class of the intersection complex of $M_{r,d}$. Our methods can be applied to the moduli spaces of semistable objects in arbitrary hereditary categories.

Theorems & Definitions (79)

• Theorem 1.1: See Theorem \ref{['main1-proof']}
• Corollary 1.2
• Theorem 1.3: See Corollary \ref{['cor-main2']}
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Example 2.4
• Example 2.5
• Lemma 2.6
• proof