Narrow Results

Let A be a complex abelian variety. The moduli space of rank one algebraic connections on A is a principal bundle over the dual abelian variety A^∨ = Pic⁰(A) for the group . Take any line bundle L on A^∨; let be the algebraic principal -bundle over A^∨ given by the sheaf of connections on L. The line bundle L produces a homomorphism . We prove that is isomorphic to the principal -bundle obtained by extending the structure group of the principal -bundle using this homomorphism given by L. We compute the ring of algebraic functions on . As an application of the above result, we show that does not admit any nonconstant algebraic function, despite the fact that it is biholomorphic to (ℂ*)^{2 dim A} implying that it has many nonconstant holomorphic functions.

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M_Sp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_Sp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where , together with the automorphisms induced by automorphisms of X.

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\mathrm{\text{Sp}}(4,ℝ)$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\mathrm{\text{Sp}}(4,ℝ)$-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

Given a compact connected Riemann surface $\mathrm{\Sigma }$ of genus $g_{\mathrm{\Sigma }}\ge 2$, and an effective divisor $D={\sum }_i{n}_i{x}_i$ on $\mathrm{\Sigma }$ with $\text{degree}(D)<2(g_{\mathrm{\Sigma }}-1)$, there is a unique cone metric on $\mathrm{\Sigma }$ of constant negative curvature $-4$ such that the cone angle at each point $x_i$ is $2\pi n_i$ [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on $\mathrm{\Sigma }$ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on $\mathrm{\Sigma }$ parametrized by a nonempty open subset of $H^0(\mathrm{\Sigma },K_{\mathrm{\Sigma }}^{\otimes 2}\otimes {{𝒪}}_{\mathrm{\Sigma }}(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case $D=0$.

Let $X=\overline{X}-D$ be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank $2$. We extend this construction to the case of any rank $r$, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.

Let $A$ be a complex abelian variety and $G$ a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal $G$-bundles and $G$-Higgs bundles on $A$. We also describe the moduli spaces of $G$-connections and the $G$-character variety for $A$.

We define a functional ${𝒥}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that ${𝒥}(h)$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating ${𝒥}(h)$ and another functional ${ℐ}(h)$, closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of ${𝒥}(h)$, which is expressed as a certain $L^2$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of ${𝒥}(h)$ if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.