Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

Step 5 of 9: Apply the decomposition theorem to the Hitchin fibration

In plain words

Pushing the topology of a space forward along a map that is not perfectly smooth, like the Hitchin fibration, can look hopelessly complicated because some fibers degenerate. The decomposition theorem, a landmark result of Beilinson, Bernstein, Deligne, and Gabber, guarantees this pushforward always breaks apart cleanly into simple, well-behaved pieces called perverse sheaves, no matter how singular the fibers are.

This hands Ngô a manageable inventory of building blocks controlling the cohomology of every fiber at once, instead of analyzing each singular fiber by hand.

Rf_* \mathbb{Q}_\ell[\dim M] \ \cong \ \bigoplus_i {}^{p}\!R^i f_* \mathbb{Q}_\ell[\dim M][-i] \qquad \text{(BBD, 1982)}

Detailed analysis

The decomposition theorem of Beilinson–Bernstein–Deligne–Gabber (BBD, 1982) states that for a proper map $f$ between algebraic varieties, the derived pushforward of the constant sheaf splits, in the derived category, as a direct sum of shifted simple perverse sheaves, each supported on some closed subvariety of the target. Applied to the Hitchin fibration $f: M \to A$ , this decomposes the cohomology of the whole family into finitely many perverse summands.

Each summand is 'attached' to a support — a subvariety of $A$ over which it is nonzero — and the entire difficulty of the fundamental lemma reduces to identifying exactly which supports occur and how large each summand is over the open, generic locus of $A$ where fibers are smooth affine Springer fibers (Hales 2011, §7; Ngô 2010, Théorème 6.4).

This is the point where the proof stops being 'find a clever integral trick' and becomes 'compute the possible supports of a decomposition in a specific geometric family' — the subject of the support theorem in the next step.

Terms in this step

perverse sheaf: A generalization of a locally constant sheaf, adapted to singular spaces, that behaves well under pushforward along algebraic maps; the natural pieces the decomposition theorem breaks a pushforward into.
decomposition theorem: A 1982 theorem of Beilinson, Bernstein, Deligne, and Gabber stating that pushing forward the constant sheaf along a proper map of algebraic varieties always yields a direct sum of simple perverse sheaves, even when the map has very singular fibers.

Knowledge used in this step

Homological algebra