Hitchin fibration - Knowlpedia

Definition

Let $X$ be a smooth projective curve, let $G$ be a reductive group with Lie algebra $\mathfrak g$ , and let $D$ be a line bundle on $X$ . A $D$ -valued $G$ -Higgs bundle is a pair $(E,\varphi)$ consisting of an algebraic principal $G$ -bundle and a section

\varphi\in H^0(X,\operatorname{ad}(E)\otimes D).

If $f_1,\ldots,f_r$ generate $k[\mathfrak g]^G$ with degrees $d_1,\ldots,d_r$ , the Hitchin fibration is

h:\operatorname{Higgs}_{G,D} \longrightarrow \mathcal A_{G,D}:= \bigoplus_{i=1}^r H^0(X,D^{\otimes d_i}), \qquad (E,\varphi)\longmapsto(f_i(\varphi))_i.

For $G=\operatorname{GL}_n$ , it records the coefficients of the characteristic polynomial of $\varphi$ .

Fibers and spectral data

Over the regular semisimple locus, a fiber is controlled by a cameral or spectral cover and a Picard stack of regular centralizers. Singular fibers encode degenerations of this abelianized geometry.

Fundamental lemma

The Hitchin fibration globalizes the local geometry of affine Springer fibers. Ngô's support theorem and comparison of the relevant endoscopic parts of its cohomology yield the fundamental lemma.

References

Nigel Hitchin, “Stable bundles and integrable systems,” Duke Mathematical Journal 54 (1987), 91–114.
Ngô Bảo Châu, “Le lemme fondamental pour les algèbres de Lie,” Publications Mathématiques de l'IHÉS 111 (2010), 1–169. Numdam.