Pierre Deligne

How to use this course

Pierre Deligne is best known for one theorem: the last and hardest of André Weil’s conjectures on the number of solutions of polynomial equations over finite fields. That proof sits at the centre of a much larger body of work. The idea of weights, which he took from Frobenius eigenvalues and transplanted to complex geometry, organizes his theory of mixed Hodge structures. His work with David Mumford created the modern theory of moduli of curves, and his work with George Lusztig created the modern representation theory of finite groups of Lie type. He received the Fields Medal in 1978 and the Abel Prize in 2013.

The technical lessons are demanding; each opens with a paragraph for the general reader, then states the main results precisely and sketches the ideas behind them. The exercises are chosen so that each can be done by hand, using the theorem of the lesson as a black box. Lessons 04 and 05 form a pair and should be read in order.

Timeline

Brussels, 1944–1964

Pierre René Deligne was born on 3 October 1944 in Etterbeek, a district of Brussels, the youngest of three children. He went to primary school in Schaerbeek from 1950. His brother, seven years older, taught him mathematics before he started school.

In September 1956 he entered the Athénée Adolphe Max, a secondary school in Brussels. There a mathematics teacher, Jeff Nijs, noticed him and lent him volumes of Nicolas Bourbaki’s Éléments de mathématique, starting with set theory. The austere, structural style of Bourbaki was his first serious mathematical reading. Nijs also introduced him to Jacques Tits, then a professor at the Université libre de Bruxelles, and while still at school Deligne began to follow Tits’s courses and seminars.

He entered the university in 1962, but his education had already moved to research level. In November 1964 Tits took him to Paris to attend the Séminaire Bourbaki and introduced him to Alexander Grothendieck. He was twenty.

Grothendieck’s school, 1965–1968

Deligne spent 1965–66 at the École Normale Supérieure in Paris and attended Grothendieck’s seminar at the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette. The seminar, recorded in the volumes known as SGA, was then building étale cohomology, the theory meant to prove the Weil conjectures (Lesson 04). Grothendieck gave him SGA material to write up and to extend. He was a collaborator on SGA 4, on the étale cohomology of schemes, and he later edited SGA 7, on monodromy, with Nicholas Katz.

He took his licence at Brussels in November 1966. From 1967 he was a junior researcher of the Belgian Fonds National de la Recherche Scientifique, on long-term leave as a guest at the IHÉS. Grothendieck treated him as an equal almost from the start. Serre, who often visited the IHÉS, became a second mentor, and the two exchanged letters for decades.

These years formed his method. He absorbed Grothendieck’s preference for the most general setting, and joined to it a gift Grothendieck lacked: a quick eye for the one concrete trick that makes a general machine produce a theorem.

Two doctorates and the IHÉS, 1968–1972

In November 1968 the Université libre de Bruxelles awarded him a doctorate for work published that year as Théorème de Lefschetz et critères de dégénérescence de suites spectrales. It showed, using the hard Lefschetz theorem, that the Leray spectral sequence of a smooth projective morphism degenerates. The argument works with any cohomology satisfying hard Lefschetz, a pattern that would recur in his later work.

The next two years were extraordinary. In a Bourbaki seminar of 1969 he attached Galois representations to modular forms of every weight at least 2 (Lesson 06). The same year, with David Mumford, he proved that the moduli space of curves of genus \(g\) is irreducible in every characteristic, introducing the notion now called a Deligne–Mumford stack (Lesson 09). In 1970 he spoke on Hodge theory at the International Congress in Nice, and in 1971 he published Théorie de Hodge II, creating mixed Hodge structures (Lesson 07).

Grothendieck left the IHÉS in 1970, and in February of that year Deligne became a permanent member, aged 25. In 1972 he received a Doctorat d’État from the Université Paris-Sud at Orsay, with Grothendieck as advisor, for his work on Hodge theory. Within a year he had proved the Weil conjectures.

The Weil conjectures

For the general reader. Count the solutions of a system of polynomial equations in a finite field with \(q\) elements, then in the field with \(q^2\) elements, \(q^3\), and so on. In 1949 André Weil predicted that these counts are governed by the shape of the corresponding complex manifold, and that they obey an analogue of the Riemann hypothesis. Proving this took 25 years and a new foundation for algebraic geometry.

Statement

Let \(X\) be a variety over \(\mathbb F_q\) and \(N_m=|X(\mathbb F_{q^m})|\). The zeta function is

Weil proved all four for curves and abelian varieties. Bernard Dwork proved rationality in 1960 by \(p\)-adic analysis. Grothendieck, with Michael Artin and Jean-Louis Verdier, built étale cohomology: for each prime \(\ell\ne p\) it gives finite-dimensional \(\mathbb Q_\ell\)-vector spaces \(H^i(\bar X,\mathbb Q_\ell)\), with the Frobenius map \(F\) acting on them, and the Lefschetz trace formula

This gives rationality, Poincaré duality gives the functional equation, and comparison with singular cohomology gives the Betti numbers. Two problems remained: that the \(P_i\) have integer coefficients independent of \(\ell\), and the Riemann hypothesis. Deligne solved both.

The proof of the Riemann hypothesis

For the general reader. Grothendieck hoped to prove Weil’s Riemann hypothesis by first establishing a set of “standard conjectures” about algebraic cycles. Those remain open. Deligne went around them. He sliced the variety into a family of lower-dimensional pieces, used the symmetries of how the pieces twist around the singular ones, and borrowed a positivity trick from the theory of modular forms.

Architecture of the proof

1. Reductions. By induction on dimension, the weak Lefschetz theorem and a product argument, it suffices to show \(|\alpha|\le q^{n/2+1/2}\) for the middle cohomology \(H^n\) of every \(X\) of even dimension \(n\). Exercise 5.1 then upgrades this to \(|\alpha|\le q^{n/2}\), and Poincaré duality gives the reverse inequality.
2. Lefschetz pencil. After blowing up, \(X\) fibres over \(\mathbb P^1\) with fibres hyperplane sections; finitely many fibres acquire a single ordinary double point. On the open set \(U\) of smooth fibres, \(R^{n-1}f_*\mathbb Q_\ell\) is a local system containing the subsystem \(\mathcal E\) spanned by the vanishing cycles, and \(H^n(X)\) is built from \(H^1_c(U,\mathcal E)\) and pieces already handled by induction.
3. Monodromy. The Picard–Lefschetz formula describes the monodromy around each singular fibre as a transvection. Since \(n-1\) is odd, cup product gives \(\mathcal E\) a symplectic form, and Kazhdan and Margulis’s theorem shows the geometric monodromy group is open in the full symplectic group. Invariant theory of \(\mathrm{Sp}\) then controls all tensor powers \(\mathcal E^{\otimes 2k}\).
4. Rankin’s trick. The symplectic invariants make the local factors of \(L(U,\mathcal E^{\otimes 2k})\) power series with nonnegative rational coefficients. A series with nonnegative coefficients converges up to its first real pole, and Grothendieck’s cohomological formula locates the poles. This bounds Frobenius eigenvalues on the stalks of \(\mathcal E\) by \(q_x^{(n-1)/2}\), which gives the bound in step 1 for \(H^1_c(U,\mathcal E)\).

The positivity idea goes back to Robert Rankin’s 1939 work on Ramanujan’s function. The proof was announced in 1973 and published as La conjecture de Weil I in the IHÉS Publications mathématiques in 1974.

Modular forms and the Ramanujan conjecture

For the general reader. In 1916 Srinivasa Ramanujan studied the coefficients \(\tau(n)\) of a remarkable power series and conjectured that \(|\tau(p)|\le 2p^{11/2}\) for every prime \(p\). Deligne showed in 1969 that this is a special case of the Weil conjectures for a certain 10-dimensional variety. His proof of the Weil conjectures five years later therefore settled Ramanujan’s conjecture.

The discriminant form is \(\Delta(z)=q\prod_{n\ge1}(1-q^n)^{24}=\sum_{n\ge1}\tau(n)q^n\), \(q=e^{2\pi iz}\), a cusp form of weight 12 for \(\mathrm{SL}_2(\mathbb Z)\). Its first coefficients are \(\tau(2)=-24\), \(\tau(3)=252\), \(\tau(5)=4830\), \(\tau(7)=-16744\). Mordell proved in 1917 that \(\tau\) is multiplicative and satisfies \(\tau(p^{r+1})=\tau(p)\tau(p^r)-p^{11}\tau(p^{r-1})\).

Deligne found \(\rho_{f,\lambda}\) in the étale cohomology of a Kuga–Sato variety, a compactified \((k-2)\)-fold fibre power of the universal elliptic curve, which is a smooth projective variety of dimension \(k-1\). The representation sits in \(H^{k-1}\), so after Weil I the eigenvalues of \(\mathrm{Frob}_p\) have absolute value \(p^{(k-1)/2}\).

With Serre in 1974, Deligne extended the construction to weight \(k=1\), where the representations have finite image and are Artin representations; the Ramanujan bound \(|a_p|\le2\) follows at once.

Mixed Hodge structures

For the general reader. The cohomology of a smooth compact complex variety splits into pieces of different “types”, by classical Hodge theory. For varieties that are singular or not compact this splitting fails. Deligne found the right replacement by imitating the finite-field picture, where Frobenius eigenvalues of different sizes, or weights, coexist in one cohomology group.

A pure Hodge structure of weight \(n\) is a finitely generated abelian group \(H_{\mathbb Z}\) with a decomposition \(H_{\mathbb C}=\bigoplus_{p+q=n}H^{p,q}\) such that \(\overline{H^{p,q}}=H^{q,p}\). The cohomology \(H^n\) of a smooth projective variety carries one. The Tate structure \(\mathbb Z(-1)\) is \(\mathbb Z\) with type \((1,1)\), weight 2.

A mixed Hodge structure is \(H_{\mathbb Z}\) with an increasing weight filtration \(W_\bullet\) on \(H_{\mathbb Q}\) and a decreasing Hodge filtration \(F^\bullet\) on \(H_{\mathbb C}\), such that \(F\) induces a pure Hodge structure of weight \(k\) on each \(\mathrm{Gr}^W_k\).

For smooth \(X\), choose a smooth compactification \(\bar X\) whose boundary \(D\) is a normal crossings divisor; the weight filtration comes from the order of the logarithmic poles along \(D\). For singular \(X\), Deligne used simplicial resolutions. Strictness is the powerful consequence: for example, it gives at once the global invariant cycle theorem, that classes on a fibre invariant under monodromy come from the total space.

Weil II and perverse sheaves

For the general reader. In 1980 Deligne turned his proof into a general theory of weights for all varieties over finite fields and for coefficients that vary in families. Two years later, with Alexander Beilinson, Joseph Bernstein and Ofer Gabber, he used it to prove the decomposition theorem for perverse sheaves, one of the deepest structural results in geometry. It was proved first over finite fields and then transferred to complex varieties.

An \(\ell\)-adic sheaf \(\mathcal F\) on \(X/\mathbb F_q\) is pure of weight \(w\) if at every closed point \(x\) the eigenvalues of \(\mathrm{Frob}_x\) on \(\mathcal F_{\bar x}\) have all complex absolute values \(q_x^{w/2}\), and mixed if it has a filtration with pure quotients.

Weil I already gave a strong bound for exponential sums. If \(f\in\mathbb F_q[x_1,\dots,x_n]\) has degree \(d\) prime to \(p\), and its top-degree part defines a smooth hypersurface in \(\mathbb P^{n-1}\), then for a nontrivial additive character \(\psi\),

Perverse sheaves. In Faisceaux pervers (1982), Beilinson, Bernstein, Deligne and Gabber defined the abelian category of perverse sheaves inside the derived category of constructible sheaves, with the intersection complexes \(IC(\bar Z,\mathcal L)\) of Goresky and MacPherson as its simple objects. They proved:

The theorem underlies the proof of the Kazhdan–Lusztig conjectures, geometric representation theory and the geometric Satake correspondence.

Stable curves and their moduli

For the general reader. All curves of genus \(g\) form a single geometric space, the moduli space \(M_g\). It is not compact, because a family of smooth curves can degenerate. In 1969 Deligne and David Mumford found the right degenerations to add, called stable curves, and used them to prove that \(M_g\) is in one piece in every characteristic. Their paper also introduced the kind of space, now called a Deligne–Mumford stack, on which moduli problems naturally live.

A stable curve of genus \(g\ge2\) is a connected projective curve of arithmetic genus \(g\) whose only singularities are ordinary double points, and in which every smooth rational component meets the rest of the curve in at least three points. Equivalently, its automorphism group is finite.

Proof idea. Over \(\mathbb C\), \(M_g\) is irreducible by classical arguments (Klein, Hurwitz, Severi) since it is dominated by a connected Hurwitz space of branched covers of \(\mathbb P^1\). The stable reduction theorem, proved using Néron models of Jacobians, gives properness. Smoothness and properness over \(\mathbb Z\), with Zariski’s connectedness principle, then carry connectedness of the complex fibre to the fibres in characteristic \(p\). Smooth and connected implies irreducible.

Deligne–Lusztig theory

For the general reader. Groups such as \(\mathrm{GL}_n(\mathbb F_q)\) are among the most important finite groups. In 1976 Deligne and George Lusztig constructed their representations geometrically, from the cohomology of certain varieties on which the group acts. This turned a collection of case-by-case computations into a uniform theory.

Let \(G\) be a connected reductive group over \(\mathbb F_q\) with Frobenius \(F\), so that \(G^F\) is a finite group. Let \(T\subseteq G\) be an \(F\)-stable maximal torus, contained in a Borel subgroup \(B=TU\) which need not be \(F\)-stable. The Deligne–Lusztig variety is

on which \(G^F\) acts on the left and \(T^F\) on the right. For a character \(\theta\) of \(T^F\), set \(R_T^\theta=\sum_i(-1)^i\,H^i_c(\tilde X,\bar{\mathbb Q}_\ell)_\theta\), a virtual representation of \(G^F\).

For \(\mathrm{SL}_2\) and the non-split torus, \(\tilde X\) is essentially the Drinfeld curve \(xy^q-x^qy=1\), whose cohomology had given Vladimir Drinfeld the discrete series of \(\mathrm{SL}_2(\mathbb F_q)\). Lusztig went on to classify all irreducible characters of finite reductive groups using this framework.

Bures and Princeton

Deligne remained a permanent member of the IHÉS from 1970 to 1984. Besides the work of Part II, the period produced the 1977 volume SGA 4½, which gave a short and self-contained route to étale cohomology and the trace formula; the theory of Shimura varieties and their canonical models (1971, 1979); his conjectures on special values of \(L\)-functions and periods (1979); and the theorem that Hodge cycles on abelian varieties are absolute Hodge cycles, published in 1982 with James Milne, Arthur Ogus and Kuang-yen Shih.

In 1984 he moved to the Institute for Advanced Study in Princeton as a professor. With George Mostow he studied monodromy of hypergeometric functions in several variables and constructed lattices in complex hyperbolic space, including non-arithmetic ones (1986, and a 1993 monograph). His 1990 paper Catégories tannakiennes characterized the categories of representations of affine group schemes intrinsically. In the late 1990s he took part, with Edward Witten and others, in a year-long IAS programme to teach quantum field theory to mathematicians, recorded in the two volumes of Quantum Fields and Strings (1999). He became professor emeritus in 2008.

His doctoral students include Lê Dũng Tráng, Miles Reid and Michael Rapoport. His influence also runs through letters: many of his long, precise letters to colleagues circulate widely, and some are posted on his IAS page.

Honours and legacy

In 1974 Deligne received the François Deruyts Prize of the Royal Academy of Belgium and the Henri Poincaré Medal of the French Academy of Sciences, and spoke at the International Congress in Vancouver. The Fields Medal followed in 1978, at the Congress in Helsinki. He received the Crafoord Prize of the Royal Swedish Academy in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, shared with Phillip Griffiths and David Mumford, and the Abel Prize in 2013. The Abel citation honoured him “for seminal contributions to algebraic geometry” and their impact on number theory, representation theory and related fields. He is a foreign associate of the French Academy of Sciences (1978) and a member of the US National Academy of Sciences, the Royal Swedish Academy and the Russian Academy of Sciences, among others. In 2006 King Albert II made him a viscount, and in 2007 Belgium issued a postage stamp in his honour.

Legacy

• Arithmetic geometry: the theory of weights, from Weil I and II, is the main tool for bounding exponential sums, counting points and proving equidistribution, and it underlies the Langlands programme over function fields.
• Hodge theory: mixed Hodge structures are standard in singularity theory, the topology of algebraic varieties and the theory of periods and motives.
• Moduli: Deligne–Mumford stacks and \(\overline M_g\) are the setting of enumerative geometry and of Gromov–Witten theory.
• Representation theory: Deligne–Lusztig theory and perverse sheaves are the foundations of geometric representation theory.

Connections

People linked to Pierre Deligne across this collection. A name without a link has no page here yet. See Pierre Deligne in the network of all 337 lives →

Teachers and advisors

• Alexander Grothendieck 1965-1972
• Jacques Tits

Students

• Lê Dũng Tráng
• Michael Rapoport
• Miles Reid

Influenced by

• André Weil
• Solomon Lefschetz

Colleagues and collaborators

• David Mumford 1969
• Dennis Sullivan 1975
• Edward Witten
• Jean-Pierre Serre 1974
• Alexander Beilinson 1982
• George Lusztig 1976
• George Mostow 1986-1993
• James Milne 1982
• Joseph Bernstein 1982
• Nicholas Katz
• Ofer Gabber 1982

Also mentioned in this life

• Emil Artin
• Vladimir Drinfeld
• Georg Frobenius
• Évariste Galois
• Carl Friedrich Gauss
• Mikhail Gromov
• Erich Hecke
• Adolf Hurwitz
• Felix Klein
• Robert Langlands
• Sophus Lie
• Grigory Margulis
• Henri Poincaré
• Bernhard Riemann
• Oscar Zariski

Fields

• algebraic geometry 17 lives
• number theory 45 lives
• representation theory 8 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Pierre Deligne, MacTutor History of Mathematics.
2. Pierre Deligne, Wikipedia.
3. Institute for Advanced Study, Pierre Deligne (CV, publications and letters).
4. Simons Foundation, Science Lives: Pierre Deligne, 2012.
5. M. Raussen and C. Skau, Interview with Pierre Deligne, Notices of the AMS 61 (2014).

Mathematics

1. P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. IHÉS 35 (1968).
2. P. Deligne, Formes modulaires et représentations \(\ell\)-adiques, Séminaire Bourbaki exp. 355 (1969), Lecture Notes in Math. 179.
3. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHÉS 36 (1969).
4. P. Deligne, Théorie de Hodge II and III, Publ. Math. IHÉS 40 (1971) and 44 (1974).
5. P. Deligne, La conjecture de Weil I, Publ. Math. IHÉS 43 (1974).
6. P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. ÉNS 7 (1974).
7. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976).
8. P. Deligne, Cohomologie étale (SGA 4½), Lecture Notes in Math. 569, Springer, 1977.
9. P. Deligne, La conjecture de Weil II, Publ. Math. IHÉS 52 (1980).
10. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982).
11. N. Katz, An overview of Deligne’s proof of the Riemann hypothesis for varieties over finite fields, Proc. Symp. Pure Math. 28 (1976).
12. J. S. Milne, Lectures on étale cohomology, course notes.
13. R. Hartshorne, Algebraic Geometry, Springer, 1977, appendix C on the Weil conjectures.