Arakelov–Milnor inequalities and maximal variations of Hodge structure

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

If S is an elliptic curve, the total space X of the cotangent bundle of S is the moduli space of rank one and degree zero Higgs bundles on S and the corresponding character variety Y is C* x C*. The punctual Hilbert scheme X^[n] of X can be identified with the moduli space of stable marked Higgs bundles on S and there is a natural isomorphism of graded vector spaces between the rational cohomology groups of the Hilbert schemes of X and Y that exchanges the perverse Leray filtration on X^[n] with the halved weight filtration on Y^[n]. We prove that there is a diffeomorphism between the Hilbert schemes that induces the given isomorphism in cohomology. We also give a complete description of Higgs bundles corresponding to subschemes of length n ≤ 3. Moreover, we discuss a conjecture by Simpson on the compactification of the moduli space of Higgs bundles and on the dual boundary complex of the character variety, proving a result going in the direction of Simpson’s conjecture.

We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively, local systems), confirming a conjecture of Boalch in these cases and extending a result of Gorsky-Nekrasov-Rubtsov. Using the McKay correspondence, we establish the autoduality conjecture for the derived categories of the moduli spaces of Higgs bundles under consideration.

Bohr–Sommerfeld Lagrangians of moduli spaces of Higgs bundles

This article is based in part on lecture notes prepared for the summer school Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles at the Institute for Mathematical Sciences at the National University of Singapore in July of 2014. The aim is to provide a brief introduction to algebraic stacks, and then to give several constructions of the moduli stack of Higgs bundles on algebraic curves. The first construction is via a bootstrap method from the algebraic stack of vector bundles on an algebraic curve. This construction is motivated in part by Nitsure's GIT construction of a projective moduli space of semi-stable Higgs bundles, and we describe the relationship between Nitsure's moduli space and the algebraic stacks constructed here. The third approach is via deformation theory, where we directly construct the stack of Higgs bundles using Artin's criterion.

Let X be a smooth projective complex curve, and let M be the moduli space of stable Higgs bundles on X (with genus g>1), with rank n and fixed determinant \xi, with n and deg(\xi) coprime. Let X' and \xi' be another such curve and line bundle, and M' the corresponding moduli space. We prove that if M and M' are isomorphic as algebraic varieties, then X and X' are isomorphic.

In this article we extend the proof given by Biswas and Gómez [Quart. J. Math. 54: 159–169, 2003] of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.

We 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.

Let X be a compact connected Riemann surface of genus g ≥ 1 equipped with a nonzero holomorphic 1-form. Let MX(r) denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g − 1) + 1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of MX(r).

Let X be a smooth projective curve over C, and let MXn,ξ be the moduli space of stable Higgs bundles on X (with genus g > 1), of rank n and fixed determinant ξ, with n and deg(ξ) coprime. Let X′ and ξ′ be another such curve and line bundle. We prove that if MXn,ξ and MX′n,ξ′ are isomorphic as algebraic varieties, then X and X′ are isomorphic.

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

In this final chapter we review certain results on stratifications of the moduli space of Higgs bundles, performed with the invariants provided by the Harder-Narasimhan filtrations. The moduli space of Higgs bundles has two stratifications. The Shatz stratification arises from the Harder-Narasimhan type of the underlying vector bundle of the Higgs bundle, and the Białynicki-Birula stratification comes from the action of the nonzero complex numbers by multiplication on the Higgs field. While these two stratifications coincide in the case of rank two Higgs bundles, this is not the case in higher rank. We thus analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles, based on results contained in [31, 98,99,100]. This relation allows some applications, as the computation of homotopy groups of the moduli space.

Let M(Spin(8,C)) be the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface X of genus g≥2. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin(8,C) acts on M(Spin(8,C)), and those Higgs bundles that admit a reduction in the structure group to G2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M(G2)→M(Spin(8,C)). In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M(Spin(8,C)). In particular, it is checked that the map M(G2)→M(Spin(8,C)) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G2 and Spin(8,C)-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

We study anti-holomorphic involutions of the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both X and G. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of X equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas–García-Prada–Hurtubise and Baraglia–Schaposnik.

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group $$G$$ . In this paper we examine the case $$G=\mathrm {SO}^*(2n)$$ . We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in $$\mathrm {SO}^*(2n)$$ .