Two-dimensional moduli spaces of rank 2 Higgs bundles overCP1with one irregular singular point

<abstract><p>We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $ 3 $ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $ 2 $ pairs, which occur in the fixed locus of this action.</p></abstract>

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INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000487 ◽

2021 ◽

pp. 1-40

Author(s):

Mirko Mauri

Keyword(s):

Riemann Surface ◽

Moduli Space ◽

Moduli Spaces ◽

Compact Riemann Surface ◽

Character Variety ◽

Degree Zero ◽

Surface Groups ◽

Higgs Bundles ◽

Character Varieties ◽

Rank 2

Abstract For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

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Moduli spaces of semistable rank-2 co-Higgs bundles over P1×P1

The Quarterly Journal of Mathematics ◽

10.1093/qmath/hax017 ◽

2017 ◽

Vol 68 (4) ◽

pp. 1139-1162 ◽

Cited By ~ 3

Author(s):

Alejandra Vicente Colmenares

Keyword(s):

Moduli Spaces ◽

Higgs Bundles ◽

Rank 2

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On the Voevodsky motive of the moduli space of Higgs bundles on a curve

Selecta Mathematica ◽

10.1007/s00029-020-00610-5 ◽

2021 ◽

Vol 27 (1) ◽

Author(s):

Victoria Hoskins ◽

Simon Pepin Lehalleur

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Characteristic Zero ◽

Rational Point ◽

Triangulated Category ◽

Higgs Bundles ◽

Projective Curve ◽

Smooth Projective Curve ◽

De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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