The Hard Lefschetz Theorem, some history and recent
progressHélène Esnault, Freie Universität Berlin

Karlsruhe, Sept. 23, 2019, DMV

Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

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Acknowledgements

Picture I took in Essen in 2010!

The new mathematical part of the lecture is based on joint work with
Moritz KERZ
I thank him for years of collaboration and for his friendship.
I thank Carlos Simpson for fruitful exchanges on this lecture.

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Acknowledgements

Picture I took in Essen in 2010!

The new mathematical part of the lecture is based on joint work with
Moritz KERZ
I thank him for years of collaboration and for his friendship.
I thank Carlos Simpson for fruitful exchanges on this lecture.
I thank Xavier Gomez-Mont, Phillip Griffiths, Susan Swartz and Alberto Verjovsky
for discussions on history.
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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton)

The son of Jewish ottoman handlers who settled down in Paris

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton)

The son of Jewish ottoman handlers who settled down in Parishe studied in ”École Centrale des Arts et Manufactures de Paris”, anengineering school .

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton)

The son of Jewish ottoman handlers who settled down in Parishe studied in ”École Centrale des Arts et Manufactures de Paris”, anengineering school .
Not being a French citizen, so not being able to access academic positions,
he emigrated to the US in 1905, where he became an engineer.

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton)

The son of Jewish ottoman handlers who settled down in Parishe studied in ”École Centrale des Arts et Manufactures de Paris”, anengineering school .
Not being a French citizen, so not being able to access academic positions,
he emigrated to the US in 1905, where he became an engineer.
He lost his both hands and a forearm in an industrial accident in 1907 andmoved towards mathematics. He became a US citizen in 1912.
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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) II
Before the end of WWII, there was almost no Jewish faculty in leading US
universities.

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) II
Before the end of WWII, there was almost no Jewish faculty in leading US
universities.
Probably under the influence of the Institute for Advanced Study (which
was then physically located at the department of Mathematics ofPrinceton U.), Princeton U. was an exception. Lefschetz became a
Princeton professor in 1924.

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) III
During WWII, many US mathematicians participated to the Manhattan
project. I seems Lefschetz did not, he ’undertook applied mathematicalwork directed at the war effort’ (Sylvia Nasar).

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) III
During WWII, many US mathematicians participated to the Manhattan
project. I seems Lefschetz did not, he ’undertook applied mathematicalwork directed at the war effort’ (Sylvia Nasar).
He went to Mexico for the first time in 1944, where anti US-feelings werestrong.

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) III
During WWII, many US mathematicians participated to the Manhattan
project. I seems Lefschetz did not, he ’undertook applied mathematicalwork directed at the war effort’ (Sylvia Nasar).
He went to Mexico for the first time in 1944, where anti US-feelings werestrong.
He became a professor at UNAM Mexico and educated until his deathgenerations of mathematicians there, notably in Algebraic Topology.

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Solomon Lefschetz 1884 (Moscow)-1972 (Princeton) III
During WWII, many US mathematicians participated to the Manhattan
project. I seems Lefschetz did not, he ’undertook applied mathematicalwork directed at the war effort’ (Sylvia Nasar).
He went to Mexico for the first time in 1944, where anti US-feelings werestrong.
He became a professor at UNAM Mexico and educated until his deathgenerations of mathematicians there, notably in Algebraic Topology.
His contribution to mathematics in Mexico is overwhelming and is still
today in the heart of many Mexican mathematicians.

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Lefschetz theory
Lefschetz had an immense influence in the areas of algebraic geometry andtopology.

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Lefschetz theory
Lefschetz had an immense influence in the areas of algebraic geometry andtopology.

If you google ’vanishing cycles’ for a nice picture, you’ll get tons of cute ...stolen bicycles!

He studied the topology of hyperplane sections of algebraic varieties,
yielding the weak Lefschetz theorems. He defined the Lefschetz pencils,
their monodromy, studied the Picard-Lefschetz formulae on vanishingcycles.

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Lefschetz theory II

His book L’analysis situs et la géométrie algébrique (1924) laid
the foundations of his theory.

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Lefschetz theory II

His book L’analysis situs et la géométrie algébrique (1924) laid
the foundations of his theory.

Wikipedia French: The name ’Analysis Situs’ hadbeen given by Leibniz to a research he led during his whole life on a symbolism
which would be to geometry what the algebraic symbolism is to numbers.
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Lefschetz theory III
The book was and still is considered by the specialists to be ’opaque’.

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Lefschetz theory III
The book was and still is considered by the specialists to be ’opaque’.
Notably, his proof of the ’hard Lefschetz theorem’, our topic today, iswrong.

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Lefschetz theory III
The book was and still is considered by the specialists to be ’opaque’.
Notably, his proof of the ’hard Lefschetz theorem’, our topic today, iswrong.
His book Topology (1930) introduced the concept Topology in place ofAnalysis Situs.

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Lefschetz theory III
The book was and still is considered by the specialists to be ’opaque’.
Notably, his proof of the ’hard Lefschetz theorem’, our topic today, iswrong.
His book Topology (1930) introduced the concept Topology in place ofAnalysis Situs.
His book Algebraic Topology (1942) introduced the concept of Algebraic
Topology in place of Combinatorial Topology.

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Lefschetz theory III
The book was and still is considered by the specialists to be ’opaque’.
Notably, his proof of the ’hard Lefschetz theorem’, our topic today, iswrong.
His book Topology (1930) introduced the concept Topology in place ofAnalysis Situs.
His book Algebraic Topology (1942) introduced the concept of Algebraic
Topology in place of Combinatorial Topology.
We owe Lefschetz the ’semantic’ we use.

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Quotations on Lefschetz
Don’t come to me with your pretty proofs. We don’t bother with thatbaby stuff around here.

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Quotations on Lefschetz
Don’t come to me with your pretty proofs. We don’t bother with thatbaby stuff around here.
He never wrote a correct proof or stated an incorrect theorem.

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Quotations on Lefschetz
Don’t come to me with your pretty proofs. We don’t bother with thatbaby stuff around here.
He never wrote a correct proof or stated an incorrect theorem.
Sylvia Nasar, A Beautiful Mind, 1998, confirmed by Phillip Griffiths.

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Some pictures: projective varieties.

In the sequel we deal with smooth projective varieties

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Some pictures: projective varieties.

In the sequel we deal with smooth projective varieties

As a first approximation, those are the zeroes of homogeneous polynomial
equations over some field. On the picture of May Ray and the next one,the field is the field of real numbers.

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Some pictures: projective varieties.

In the sequel we deal with smooth projective varieties

As a first approximation, those are the zeroes of homogeneous polynomial
equations over some field. On the picture of May Ray and the next one,the field is the field of real numbers.
even if the set of zeroes of polynomial equations isempty, the mere fact that we have equations entitles us to talk onvarieties. The second approximation is that we have in addition to characterize the smoothness notion,
which essentially means (over fields like the complex numbers) that if we caress thegeometric object with our fingers, it never pricks.

The first approximation here is that

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Hard Lefschetz in complex geometry

Theorem (Formulated by Lefschetz 1924, proved by Hodge 1941)
Let X be a complex projective manifold of dimension d, and η be theChern class in Betti cohomology H 2 (X , Z) of an ample line bundle. Thenfor all i ∈ N, the cup-product map∪η i : H d−i (X , Q) → H d+i (X , Q)
is an isomorphism.

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Hard Lefschetz in complex geometry II
It is a structural theorem which enables one to write cohomology as adirect sum of ’primitive cohomologies’ (called Lefschetz decomposition). It
has immense consequences.

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Hard Lefschetz in complex geometry: Hodge theory
Hodge (1941) introduced Harmonic Theory on the space of differential
forms in the study of de Rham cohomology of complex manifolds.

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Hard Lefschetz in complex geometry: Hodge theory
Hodge (1941) introduced Harmonic Theory on the space of differential
forms in the study of de Rham cohomology of complex manifolds.
This contains the notion of orthogonality and enables one to deduce thatthe intersection pairing restricted to the space of Lefschetz’ vanishingcycles is non-degenerated, which is essentially equivalent to Hard
Lefschetz.

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Hard Lefschetz in complex geometry: Hodge theory
Hodge (1941) introduced Harmonic Theory on the space of differential
forms in the study of de Rham cohomology of complex manifolds.
This contains the notion of orthogonality and enables one to deduce thatthe intersection pairing restricted to the space of Lefschetz’ vanishingcycles is non-degenerated, which is essentially equivalent to Hard
Lefschetz.
In particular, Hard Lefschetz holds more generally for Kähler manifolds, so
compact complex manifolds endowed with a Kähler form. Its classη ∈ H 2 (X , R) is then the one in the theorem.

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Hard Lefschetz in complex geometry: Hodge theory
Hodge (1941) introduced Harmonic Theory on the space of differential
forms in the study of de Rham cohomology of complex manifolds.
This contains the notion of orthogonality and enables one to deduce thatthe intersection pairing restricted to the space of Lefschetz’ vanishingcycles is non-degenerated, which is essentially equivalent to Hard
Lefschetz.
In particular, Hard Lefschetz holds more generally for Kähler manifolds, so
compact complex manifolds endowed with a Kähler form. Its classη ∈ H 2 (X , R) is then the one in the theorem.
Hard Lefschetz is one central piece of Hodge theory which controls
cohomology of compact Kähler manifolds, e.g. of smooth complex
projective manifolds.
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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson)

In 1992 Simpson defined the notion of non-abelian Hodge theory onsmooth complex projective varieties.

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson)IILet ρ : π1top (X , x) → GLr (C) be a representation of the topologicalfundamental group.

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson)IILet ρ : π1top (X , x) → GLr (C) be a representation of the topologicalfundamental group.
It yields a local system V of rank r on X .

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson)IILet ρ : π1top (X , x) → GLr (C) be a representation of the topologicalfundamental group.
It yields a local system V of rank r on X .
It is a topological bundle on X , defined as the quotient of the productXx × Cr by the diagonal action of π1top (X , x). Here Xx → X is theuniversal cover associated to the choice of x, π1top (X , x) acts on Xx as thegroup of deck transformations and on Cr via ρ.

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson)IILet ρ : π1top (X , x) → GLr (C) be a representation of the topologicalfundamental group.
It yields a local system V of rank r on X .
It is a topological bundle on X , defined as the quotient of the productXx × Cr by the diagonal action of π1top (X , x). Here Xx → X is theuniversal cover associated to the choice of x, π1top (X , x) acts on Xx as thegroup of deck transformations and on Cr via ρ.
Then non-abelian refers to the replacement of X by (X , V).

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson) III
Under the assumption that ρ is semi-simple, Simpson showed the existenceof a harmonic metric on V made a C ∞ bundle. This enabled him todevelop a Hodge theory with this metric and notably to prove the
non-abelian Hard Lefschetz.

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Hard Lefschetz in complex geometry: non-abelian Hodge
theory (Simpson) III
Under the assumption that ρ is semi-simple, Simpson showed the existenceof a harmonic metric on V made a C ∞ bundle. This enabled him todevelop a Hodge theory with this metric and notably to prove the
non-abelian Hard Lefschetz.

Theorem (Simpson 1992)
Let X be a compact Kähler manifold of dimension d, V be a semi-simple
local system of rank r , and η be the Chern class in Betti cohomology
H 2 (X , C) of a Kähler class. Then for all i ∈ N, the cup-product map∪η i : H d−i (X , V) → H d+i (X , V)
is an isomorphism.
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Examples
Example (Finite monodromy)
Of course, if V has finite monodromy, we can replace X by a finite
topological cover Y → X on which V becomes a trivial local system, apply
Hard Lefschetz on Y , and descend the information by a trace argument.
However, finite monodromy local systems are not often.

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Examples II
Example (Rank 1)
e.g. if r = 1, we understand well their moduli space: it contains a complex
torus as a finite index subgroup. A complex torus is like on the picture: acircle (on the top) cross the (strictly positive, here from top to bottom to

infinity) line.

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Examples II
Example (Rank 1)
e.g. if r = 1, we understand well their moduli space: it contains a complex
torus as a finite index subgroup. A complex torus is like on the picture: acircle (on the top) cross the (strictly positive, here from top to bottom to

infinity) line.

ExampleThe points which correspond to torsion (i.e. with finite monodromy) rank
1 local systems are all located on one horizontal section. So they are notdense in the real topology. It makes it difficult to deduce Simpson’s
theorem from the standard Hard Lefschetz. We remark however already here for later discussionthat they are Zariski dense.
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The Riemann hypothesis

The Riemann hypothesis (1859) predicts that for s ∈ C , the Riemannzeta function11ξ(s) = 1 + s + s + . . . ,23which converges for Re(s) > 1, has all but its trivial zeroes −2, −4, . . .
located on the vertical line Re(s) = 21 .

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The Weil conjectures

Weil (1948) (see e.g. a Lecture I have at the Bibliothèque Nationale de France, which is linked on my webpage)defined the zeta function of a smooth projective variety X defined over afinite field Fq , by counting the number of Fqn -rational points of X as ngoes to infinity.

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The Weil conjectures

Weil (1948) (see e.g. a Lecture I have at the Bibliothèque Nationale de France, which is linked on my webpage)defined the zeta function of a smooth projective variety X defined over afinite field Fq , by counting the number of Fqn -rational points of X as ngoes to infinity.
He predicted the existence of a cohomology theory H i (X ) on which the
Frobenius F acts, which would express the zeta function as the trace of
the Frobenius acting on cohomology.

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The Grothendieck school in relation to the Weil conjectures`-adic étale cohomology theory has been developed by the Grothendieckschool in the 60s, 70s. Dwork (1960), independently of the Grothendieckschool, and then Grothendieck (1965) proved the necessary properties ofthe zeta function.

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Grothendieck-Lefschetz trace formula
In summary, for X0 a smooth projective variety of dimension d definedover the finite field Fq , X its base change to an algebraic closure Fq , andF the ’geometric’ Frobenius acting on `-adic cohomology H i (X ), one has
the Grothendieck-Lefschetz trace formula|X0 (Fq )| = Trace(F |H • (X )) =2d
X(−1)i Trace(F |H i (X ))i=0Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

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The Weil conjectures II: Deligne
Weil, in analogy to the Riemann hypothesis, predicted that read via
complex embeddings, the absolute value of the eigenvalues of the
Frobenius acting on H i depends only on i and are different for different i.

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The Weil conjectures II: Deligne
Weil, in analogy to the Riemann hypothesis, predicted that read via
complex embeddings, the absolute value of the eigenvalues of the
Frobenius acting on H i depends only on i and are different for different i.
One says that the cohomology is pure, or that the weights of thecohomology are pure, and are different for different i.

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The Weil conjectures II: Deligne
Weil, in analogy to the Riemann hypothesis, predicted that read via
complex embeddings, the absolute value of the eigenvalues of the
Frobenius acting on H i depends only on i and are different for different i.
One says that the cohomology is pure, or that the weights of thecohomology are pure, and are different for different i.

In 1973 Deligne proves the Weil conjectures. In 1980 he publishes a moreconceptual proof.
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The Weil conjectures and Hard Lefschetz (Deligne)
Notably he proves the Hard Lefschetz theorem.

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The Weil conjectures and Hard Lefschetz (Deligne)
Notably he proves the Hard Lefschetz theorem.

Theorem (Deligne 1980)
Let X0 be a smooth projective variety of dimension d defined over a finite
field Fq , let X be its base change to an algebraic closure Fp . Let η be theChern class in `-adic cohomology H 2 (X , Q` ) of an ample line bundle.
Then for all i ∈ N, the cup-product map∪η i : H d−i (X , Q` ) → H d+i (X , Q` )is an isomorphism.

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The Weil conjectures and Hard Lefschetz (Deligne) II
What replaces Harmonic Theory in Hodge theory on the complex side is
Deligne’s theory of weights for varieties defined over a finite field.

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The Weil conjectures and Hard Lefschetz (Deligne) II
What replaces Harmonic Theory in Hodge theory on the complex side is
Deligne’s theory of weights for varieties defined over a finite field.
It enables him to conclude that the intersection pairing restricted to thespace of Lefschetz’ vanishing cycles is non-degenerated.

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Examples
Example (Finite monodromy)
Of course as we saw on the complex side, we could replace Q` coefficientsby any `-adic local system with finite monodromy V. Such a V isnecessarily defined over the base change of X0 to a finite extension of Fq .
We would apply Deligne’s Hard Lefschetz on the pull-back of V to a finiteétale cover Y → X which kills the monodromy, and then descend via a
trace argument.

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Examples
Example (Finite monodromy)
Of course as we saw on the complex side, we could replace Q` coefficientsby any `-adic local system with finite monodromy V. Such a V isnecessarily defined over the base change of X0 to a finite extension of Fq .
We would apply Deligne’s Hard Lefschetz on the pull-back of V to a finiteétale cover Y → X which kills the monodromy, and then descend via a
trace argument.

Example (Rank 1)
For example in rank r = 1. However, most `-adic local systems are notdefinable over the base change of X0 to a finite extension of Fq . The aimof the last part of the lecture is to explain that we can make more precise
this most. This leads to a proof of a new case of Hard Lefschetz for which
we do not have Deligne’s theory of weights at disposal.
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Zariski density of torsion points.
Theorem (E-Kerz 2019)
Let X be a smooth projective variety defined over an algebraically closed
field of characteristic p > 0.
There is a noetherian space G over Q` , such that its points G(Q` )form a subgroup of discrete index in the group of rank 1 `-adic local
systems.
With respect to the Zariski topology stemming from thisnoetherianity, the locus S◦ of points which violate Hard Lefschetz is
Zariski constructible.
The Zariski closure of S◦ , if not empty, necessarily is the
Zariski closure of its torsion points.

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Hard Lefschetz in rank 1 is positive characteristic.
Corollary (E-Kerz 2019)
Let X be a smooth projective variety defined over an algebraically closed
field of characteristic p > 0, let V be a rank 1 local system. Let η be theChern class in `-adic cohomology H 2 (X , Q` ) of an ample line bundle.
Then for all i ∈ N, the cup-product map∪η i : H d−i (X , V) → H d+i (X , V)
is an isomorphism.

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Logic of proof: Drinfeld

The logic of proof, as exposed, relies on Drinfeld’s work: Indeed, if not
empty, S◦ must contain torsion points, which violates Deligne’s Hard
Lefschetz.

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Proof
The proof itself relies on `-adic geometry, using methods developed
originally for p-adic geometry. It uses Deligne’s purity theorem, Tate’s
semi-simplicity theorem.

Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

Hard Lefschetz

30 /
30

Proof
The proof itself relies on `-adic geometry, using methods developed
originally for p-adic geometry. It uses Deligne’s purity theorem, Tate’s
semi-simplicity theorem.
It is the profinite analog of Simpson’s theorem in complex geometry
describing closed subloci of the moduli of rank 1 complex local systems
which viewed in de Rham theory are algebraic (bialgebraic):

Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

Hard Lefschetz

30 /
30

Proof
The proof itself relies on `-adic geometry, using methods developed
originally for p-adic geometry. It uses Deligne’s purity theorem, Tate’s
semi-simplicity theorem.
It is the profinite analog of Simpson’s theorem in complex geometry
describing closed subloci of the moduli of rank 1 complex local systems
which viewed in de Rham theory are algebraic (bialgebraic):
They have the property that they are the Zariski closure of their torsion
points.

Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

Hard Lefschetz

30 /
30

Proof
The proof itself relies on `-adic geometry, using methods developed
originally for p-adic geometry. It uses Deligne’s purity theorem, Tate’s
semi-simplicity theorem.
It is the profinite analog of Simpson’s theorem in complex geometry
describing closed subloci of the moduli of rank 1 complex local systems
which viewed in de Rham theory are algebraic (bialgebraic):
They have the property that they are the Zariski closure of their torsion
points.

Thank you for your attention.

Karlsruhe, Sept. 23, 2019, DMVHélène Esnault, Freie Universität Berlin

Hard Lefschetz

30 /30