This paper gives a self-contained, detailed account of the construction and compactification of the moduli space of Higgs bundle on (families of) curves. It is divided into five parts: I. The theorem of the cube; II. $G$-bundles; III. Abelianisation; IV. Projective connections; V. Infinitesimal parabolic structure.

Part I starts with a theorem on the determinant of the cohomology of coherent sheaves on a curve. It is shown that the theorem of the cube follows from this result. A second application is at the basis of the construction, via theta-functions, of global sections of determinant bundles on the moduli space of Higgs bundles. More precisely, let $S$ be a noetherian base scheme for the family of curves $\pi :C\to S$, with $\pi $ proper, all fibers of dimension $\le 1$, and such that ${\pi}_{*}\left({\mathcal{O}}_{C}\right)={\mathcal{O}}_{S}$. A Higgs bundle on $C$ is a vector bundle $\mathcal{F}$ together with a section $\theta $ of ${\Gamma}(C,\mathcal{E}nd\left(\mathcal{F}\right)\otimes {\omega}_{C})$. The coefficients of the characteristic polynomial of $\theta $ define global sections ${f}_{i}\in {\Gamma}(C,{\omega}^{i})$, and the affine space classifying such sections is called the characteristic variety $\mathcal{C}har$ (it depends on $\text{rk}\left(\mathcal{F}\right))$, and $\mathcal{F}$ defines a point $\text{char}\left(\mathcal{F}\right)$ in $\mathcal{C}har$. Such a Higgs bundle will often be denoted $(\mathcal{F},\theta )$. One has the notion of (semi)-stability for Higgs bundles, and any semistable Higgs bundle admits a Jordan- HĂ¶lder (JH) filtration by subbundles with stable quotieents of constant ratio degree/rank. The isomorphism classes and multiplicities of these stable components are independent of the filtration. Two semi-stable Higgs bundles are called JH-equivalent if these coincide. A result on JH- equivalence is derived and used to show that the theta-functions separate points in the moduli space of Higgs bundles. The moduli-space of stable Higgs bundles of given rank and degree is constructed as an algebraic space ${\mathcal{M}}_{\theta}^{0}$. Then ${\mathcal{M}}_{\theta}^{0}$ embeds as an open subscheme into the onrmalization ${\mathcal{M}}_{\theta}$ of ${\mathbb{P}}^{N}\times \mathcal{C}har$ (for suitable $N)$ in ${\mathcal{M}}_{\theta}^{0}$.

In part II one considers a reductive connected algebraic group $\mathcal{G}$ over a smooth projective connected curve $C$ over a field $k$. A $\mathcal{G}$-torsor $P$ on $C$, together with an element

is called semistable if $(\text{Lie}\left({\mathcal{G}}_{P}\right)$, ad$\left(\theta \right))$ is a semistable Higgs bundle of degree zero. One also has the notion of stable $P$. The main result on semistable pairs $(P,\theta )$ is the following semistable reduction theorem: If $V$ is a complete discrete valuation ring with fraction field $K$, $C\to V$ a smooth projective curve, $({P}_{K},{\theta}_{K})$ a semistable pair (associated with a connected reductive group $\mathcal{G}$ over $C)$ whose characteristic is integral over $V$, then there exists a finite extension ${V}^{\text{'}}$ of $V$ such that the base extension of $({P}_{K},{\theta}_{K})$ extends to a semistable pair on ${C}_{{V}^{\text{'}}}$. Furthermore, if the special fiber of this extension is stable, then any other semistable extension is isomorphic to it. For stable $(P,\theta )$ one is led to construct an algebraic moduli stack ${\mathcal{M}}_{\theta}^{0}\left(\mathcal{G}\right)$ and the coarse moduli space ${M}_{\theta}^{0}\left(\mathcal{G}\right)$ which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a ${M}_{\theta}\left(\mathcal{G}\right)$ as the normalisation of a ${\mathbb{P}}^{N}$ in ${M}_{\theta}^{0}\left(\mathcal{G}\right)$. Then ${M}_{\theta}\left(\mathcal{G}\right)$ is projective over $\mathcal{C}har$ and contains ${M}_{\theta}^{0}\left(\mathcal{G}\right)$ as an open subscheme. Then, for example, if $C$ has genus $>2$, the boundary ${M}_{\theta}\left(\mathcal{G}\right)-{M}_{\theta}^{0}\left(\mathcal{G}\right)$ has codimension $\ge 4$. Many other results are derived.

In part III the theory is extended to exceptional groups. As a corollary of the theory one obtains, with the notations above, that the set of connected components of the moduli space ${\mathcal{M}}^{0}\left(\mathcal{G}\right)$ of stable (Higgs) $\mathcal{G}$-bundles coincides with that of ${\mathcal{M}}_{\theta}^{0}\left(\mathcal{G}\right)$, ${M}_{\theta}\left(\mathcal{G}\right)$ as well as that of a generic fiber of ${\mathcal{M}}_{\theta}^{0}\left(\mathcal{G}\right)\to \mathcal{C}har$, under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of ${\mathcal{M}}_{\theta}^{0}\left(\mathcal{G}\right)$, all global functions are obtained by pullback from $\mathcal{C}har$.

In part IV the accent is on ${\mathcal{M}}^{0}\left(\mathcal{G}\right)$, where $\mathcal{G}$ is the twisted form of some semi-simple $G$. The notion of ${{\Omega}}_{C}$- connections $\nabla $ on $\mathcal{G}$-torsors $P$ is introduced. ${\mathcal{M}}_{\nabla}^{0}\left(\mathcal{G}\right)$ denotes the moduli stack of such pairs $(P,\nabla )$ with $P$ stable. It is fibered over ${\mathcal{M}}^{0}\left(\mathcal{G}\right)$. Over $\u2102$, ${\mathcal{M}}_{\nabla}^{0}\left(\mathcal{G}\right)$ classifies bundles with integrable connections, i.e. representations of ${\pi}_{1}\left(C\right)$. A locally faithful $\mathcal{G}$-representation $\mathcal{F}$ defines a line bundle $\mathcal{L}=\mathcal{L}\left(\mathcal{F}\right)$ on ${\mathcal{M}}^{0}\left(\mathcal{G}\right)$. Then the pullback of $\mathcal{L}$ to ${\mathcal{M}}_{\nabla}^{0}\left(\mathcal{G}\right)$ has a connection $\nabla $. Its curvature can be described explicitly.

The final part V discusses parabolic structures in the sense of C. Seshadri. The parabolic analogue of a Higgs bundle is introduced and a theory parallel to the one in the foregoing parts is sketched.

##### MSC:

14H10 | Families, algebraic moduli (curves) |

14H60 | Vector bundles on curves and their moduli |

14F05 | Sheaves, derived categories of sheaves, etc. |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) |