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

Step 6 of 9: Ngô's support theorem: no support is smaller than the whole base

In plain words

A priori, some of the building-block pieces from the decomposition theorem could be concentrated on a small, exceptional part of the parameter space $A$ — say, only over points where the fiber is unusually singular. Ngô's support theorem rules this out: every relevant piece is spread across all of $A$ , as large and uniform as it could possibly be.

Intuitively, the family cannot 'hide' extra topology in special corners; whatever happens generically also happens everywhere, in a precisely controlled sense.

\mathrm{Supp}(\text{perverse summand}) = A \qquad \text{(no smaller support occurs)}

Detailed analysis

Ngô's support theorem (2010, Théorème 6.4, building on partial results of Goresky, Kottwitz, and MacPherson for classical groups) shows that the supports of the nonzero perverse summands from the decomposition theorem cannot be proper closed subvarieties of $A$ : each summand's support must be a union of components of full dimension. The proof combines a dimension estimate on possible supports — via a ' $\delta$ -regularity' bound on the discriminant, roughly measuring how singular the generic fiber over a locus is — with a global argument using an ample class on the Hitchin moduli space and the hard Lefschetz theorem.

This is the single deepest new geometric input of Ngô's paper; it is what pins down the cohomology of the Hitchin fibration precisely enough to compare the $G$ -side and the $H$ -side, rather than merely bounding it.

With the support pinned down to be everywhere, the cohomology of the fibration is determined, on the dense open locus where fibers are honest affine Springer fibers, by exactly the objects from Step 3.

Terms in this step

support theorem: Ngô's 2010 theorem asserting that every perverse summand in the decomposition of the Hitchin fibration's pushforward is supported on the whole parameter space $A$ (up to full-dimensional pieces), ruling out contributions hidden on smaller subvarieties.
$\delta$ -regularity: A technical condition measuring how the discriminant (recording how singular the centralizer of $\gamma$ is) grows across the Hitchin base $A$ ; used to bound the dimension of possible supports.

Knowledge used in this step

Homological algebra Algebraic geometry