Intersection Cohomology of the Moduli Space of Higgs Bundles on a Genus 2 Curve

Abstract

Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

P=W conjectures for character varieties with symplectic resolution

“…) we haveIP t (M ) = P t ( M ) − P t (Σ)t 2 − P t (Ω)t 6 = (1 + 2t 2 + 23t 4 + 34t 6 ) − (1 + 6t 2 + 1)t 2 − 16t 6 = 1 + t 2 + 17t 4 + 17t 6 ;see also[26, Th. 6.1].Since the differentials of the local-to-global spectral sequence(26) are Γ-equiva-* ( N ) and H * (Σ) H * (S) ⊂ H * (S + ), and as the regular representation on the 16-dimensional vector spaces 16 j=1…”

“…-The spectral sequence (26) degenerates at the first page, and the Poincaré polynomial P t (X) := 2 dim X k=0 (−1) n dim H k (X, Q) can be written The Hitchin map χ :…”

Intersection cohomology of character varieties for punctured Riemann surfaces

“…As they are singular it is also interesting to consider their intersection cohomology. Felisetti [Fel21] computed the intersection cohomology in rank n = 2 and genus g = 2. Mauri [Mau21a] generalized the computation to rank n = 2 and arbitrary genus.…”

Intersection cohomology of character varieties for punctured Riemann surfaces

“…In the degree zero case, which corresponds to non-generic monodromy, the moduli space is singular. In this situation intersection cohomology is computed by Felisetti [Fel21] for rank n = 2 and genus g = 2. García-Prada, Heinloth, Schmitt [GPHS11] gave a recursive algorithm to compute the motive of the Dolbeault moduli space.…”