# Why is the hard Lefschetz theorem “hard”?

Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear morphism between cohomology groups

$$L^k : H^{n-k}(X,\mathbb C) \longrightarrow H^{n+k}(X,\mathbb C)$$

which is simply given by cup product against the class $[\omega]^k$. The hard Lefschetz theorem says that this is in fact an isomorphism of vector spaces.

Question: Why do we call this the "hard" Lefschetz theorem?

Modern proofs of this theorem are not that involved; one picks a Kähler metric $\omega$ and proves the Kähler identities on $X$, and the rest then follows from the existence of primitive decompositions. Thus it seems a bit of hype to call the theorem "hard".

One might think this is to distinguish this from another theorem of Lefschetz, often called the "weak" Lefschetz theorem, which gives a similar result in the case where $[\omega]$ is the Chern class of an ample line bundle. But then we'd surely call this the "strong" Lefschetz theorem, right?

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According to Wikipedia (which, thankfully, is still working), "This is the hard Lefschetz theorem, christened in French by Grothendieck more colloquially as the Théorème de Lefschetz vache." –  Isaac Solomon Jan 17 '12 at 15:41
I did see that Wikipedia article, but I'm unclear on: 1) was this theorem known as the Hard Lefschetz theorem before this and did Grothendieck just give it a slangy name in French, 2) if not, then why did Grothendieck think it was "vache"? –  Gunnar Magnusson Jan 17 '12 at 18:16
According to every source I can find, "vache" is French for "cow." Anyone want to explain what's going on there? –  Daniel McLaury Dec 23 '13 at 7:36
@DanielMcLaury: It does mean "cow", but as an adjective or adverb it's also slang for something unexpected and bad, something like the (British) "nasty", "(surprisingly) difficult" or "fearsome". If a friend shows you some awful work he has to do you can exclaim "Oh, la vache!" in sympathy. See also en.wikibooks.org/wiki/French/Appendices/Slang –  Gunnar Magnusson Dec 23 '13 at 10:01

This question can only have a subjective answer (which is actually fun, from time to time!), so here are a few personal remarks.

1) You are a dynamic PhD student working in 2012 under the supervision of Demailly, a world leader in complex algebraic geometry.
You have at your disposal a technology that didn't exist on Lefschetz' time: singular and De Rham cohomology, higher homotopy groups, Kähler manifolds, Hodge theory,...
Even the abstract notion of a finite-dimensional vector space had not been axiomatized.
So when you claim that the theorem is not that hard, you should not lose sight of the historic context in which Lefschetz "proved" his theorem in 1924.

2) I wrote "proved" in quotes, since as Sabbah diplomaticallty puts it, Lefschetz' proof was "insufficient".
So the theorem was not easy, even for Lefschetz.

3) The theorem has fascinated many Fields medalists and other giants who gave proofs of some version of the theorem: Andreotti, Frankel, Thom, Bott, Kodaira, Spencer, Artin, Grothendieck, Deligne.
This is certainly an indication of the depth of the theorem...

4) Like you I am enthusiastic about complex algebraic manifolds and am grateful for the transcendental methods , like Kähler theory, which allow us to study them.
However algebraic geometers also want to consider algebraic varieties in characteristic $p$, and there these transcendental tools unfortunately completely break down.
Hard Lefschetz for smooth varieties over finite fields was proved by Deligne only in 1980, after much preliminary work by himself and Grothendieck (cf. SGA7).
I would surmise that the terminology "Lefschetz vache" introduced by Grothendieck is to be understood in that context.

5) Finally even in the complex case, I find the proof of hard Lefschetz starting from scratch not so easy.
I'll let you and the other users judge by linking to a free online course of Sabbah on Hodge theory and hard Lefschetz (in the Introduction of which he writes the diplomatic remark mentioned above!)

Edit
Since this is a good-humoured, non-technical answer, I'll take the liberty of quoting the following picturesque metaphor by Lefschetz:

It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry

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You are right about many things. It's true that proofs that exist after decades of rewriting are very streamlined and sometimes make it difficult to see where the original lions dwell, especially when combined with the power of modern technology. I do wish I had a better view of the difficulties Lefschetz, Grothendieck and others faced, but people tend to write and talk about things that worked out rather than those that did not. PS: that quote by Lefschetz is pure gold. –  Gunnar Magnusson Jan 18 '12 at 19:44