# Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry.

Can anyone explain why this is ?

• A while back I found this video of the lecture "Hodge Theory -- From Abel to Deligne" by Phillip Griffiths" youtube.com/watch?v=hPyDz5R5YaY rather enlightening. – j.c. Oct 12 at 16:49

First of all, the results are miraculous. Deligne's Hodge II and Hodge III give just a few example applications of the kind of results you can prove using mixed Hodge theory; these are great theorems which just fall out of the general theory. People quickly figured out more applications, like the Hodge-Deligne polynomial, which is itself a complete miracle - and its existence just falls out of the general theory.

But there is also the fact that the results seemingly came out of nowhere. In fact Deligne was motivated by the philosophy of motives, which was not very well known/understood at the time outside the initiated few around Grothendieck. To my knowledge the only accounts of motives in the literature at the time were Kleiman's paper in the Oslo proceedings and Manin's paper on the blow-up formula for Chow motives, and it would be a long time until anyone talked about "mixed motives". The weight filtration appeared naturally in the $$\ell$$-adic cohomology when you considered a variety which was not necessarily smooth and projective, and the Frobenius eigenvalues on $$\mathrm{Gr}^W_i H^k$$ would be of absolute value $$q^{i/2}$$, so they would look like part of the degree $$i$$ cohomology of a smooth proper variety. And "by motivic philosophy" that phenomenon should correspond to something on the Hodge theory side. But without this motivic philosophy as guidance the weight filtration looks completely ad hoc and unmotivated.

• I think the (great) idea that "mixed motives", hence everything else, should have a weight filtration whose quotients are pure is probably an idea of Deligne's. At least, I've never seen anything like it in Grothendieck's writings. [I would be interested in the comments of others on this.] – anon Oct 13 at 12:31
• @anon That's my impression as well. I didn't mean to imply otherwise. – Dan Petersen Oct 13 at 13:35
• Naively, the Frobenius eigenspace decomposition suggests that the structure of weights should be formulated as a grading on cohomology. The fact that Deligne formulated it as filtration, which gives finer structure, showed remarkable insight. And the fact that Deligne was able to get this formulation to work in the Hodge setting (lemma on two filtrations...) showed remarkable technical virtuosity. – Donu Arapura Oct 13 at 13:40
• @Donu Thanks! The Frobenius eigenspace decomposition only splits the weight filtration over a finite field, though. Over (say) a number field, where there are infinitely many (non-commuting) Frobenii to diagonalize rather than a single one, it turns out that only the filtration is canonical. And one reason the filtration is canonical is that it can also be built from a dévissage argument like those used in Hodge II and III. Moreover, for some reason, no matter how you try to build your cohomology out of smooth projective things, the filtration always goes "in the same direction". – Dan Petersen Oct 13 at 14:15
• ...so I imagine that's how the idea came to him. PS I know you know all this Donu - my comment is also a supplement. – Dan Petersen Oct 13 at 14:15

The laudatio for the Wolf prize explains it like this:

Central to modern algebraic geometry is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was traditionally mysterious and problematic. In critical special cases, i.e., curves, it made sense, i.e., the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, there was some sort of structure locally, but globally everything remained mysterious. [...] Building on Mumford’s and Griffiths’ work, Pierre Deligne demonstrated how to extend the variation of Hodge theory to singular varieties. This advance, called mixed Hodge theory, allowed explicit calculation on the singular compactification of moduli spaces that came up in Mumford’s geometric invariant theory.