Spin(8, ℂ)-Higgs bundles fixed points through spectral data

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.

In this research, we study the geometry of the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of Spin(8,C). We establish a characterization of triality invariance, proving that a Spin(8,C)-Higgs bundle admits a reduction to the exceptional group G2 if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting G2-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from G2-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset.

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

Hitchin fibration under ramified coverings

Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}_{FH}(G)$ to the moduli space $\mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, \{H_x\}_{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, \{\text{I}_{r\times r}\}_{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}_{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

Let X denote a compact Riemann surface of genus g > 1. At each point P~X, there is a sequence of g integers, l=71(P) g. Every compact Riemann surface of genus > 1 has a finite number of Weierstrass points. (See [6] for details.) The question of describing the moduli of compact Riemann surfaces of genus g which have a Weierstrass point with a specified first nongap has been answered by Rauch [14] and Arbarello [1]. The problem of describing the moduli of curves of genus g which have a Weierstrass point with a specified gap sequence has been considered by Hensel-Landsberg [7] and Rauch [15] and recently results on this question have been obtained by Pinkham [13] and RimVitulli [16]. In [9], we defined complex spaces of Weierstrass points of the universal curve and obtained a theorem concerning the smoothness and dimension of these spaces. From this, one can recover the moduli results of Rauch [14]. In order to study the higher Weierstrass nongaps and the entire gap sequence, we must extend our definition in [9]. We do this in w and then prove a theorem concerning the smoothness and dimension of these spaces which is analogous to the main result in [9]. F rom this, we can recover a moduli result of Pinkham [13]. In w 3, we study the geometry of these complex spaces of Weierstrass points of the universal curve of genus 4. We obtain a description of the moduli of Teichmiiller surfaces of genus 4 which have a Weierstrass point with a specified gap sequence. In the following, a complex space is not necessarily reduced. If Y is a complex space, then IYI will denote the underlying set of points. If Y, Z1 . . . . . Z , are closed complex subspaces of a complex space, then by Y Z 1 Z 2 . . . Z ,

The Uniformization Theorem due to Koebe and Poincare implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a companion paper "Moduli spaces of holomorphic triples over compact Riemann surfaces" (math.AG/0211428) we prove that these moduli spaces of triples are non-empty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p,q)-representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor-Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is non-empty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

Let be an arbitrary non‐compact hyperbolic Riemann surface, that is, not the complex plane or punctured plane. Let be , or . We show that, on a Higgs bundle in the ‐Hitchin section over , there always exists a harmonic metric satisfying certain nice properties. Moreover, when the Higgs field has bounded spectral curve with respect to the complete hyperbolic metric on , such a harmonic metric is unique. As a result, we obtain a new proof of the existence and uniqueness of a harmonic metric for Higgs bundles in the ‐Hitchin section over a compact hyperbolic Riemann surface. Moreover, we show an existence result of harmonic metrics for a more general family of Higgs bundles that admit a full holomorphic filtration.

We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higgs bundles.

We prove that a hyperelliptic Riemann surface of genus $g$ can be completely conformally and minimally immersed in ${R^3}$ with finite total curvature with at most $3g + 2$ punctures; an arbitrary compact Riemann surface of genus $g$ can be so immersed with at most $4g$ punctures. Moreover, we show that there is at least a one-parameter family of nonisometric such immersions for a given punctured compact Riemann surface. Our results improve earlier results of Gackstatter-Kunert and the author.

Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure that is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold that is called here as ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${\mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.

Sixty years ago Hirzebruch observed how the vanishing of the Stiefel–Whitney class w 2 led to integrality of the $$\hat{A}$$ -genus of an algebraic variety [Hirz1]. This was one motivation for the Atiyah–Singer index theorem but also for my own thesis about Dirac operators and Kähler manifolds. Indeed the interaction between topology and algebraic geometry which he developed has been a constant theme in virtually all my work.

departmental bulletin paper