#### Translator’s note

Deligne, P. “Théorie de Hodge I.” Actes du Congrès intern. math. 1 (1970), 425–430. [PDF].

The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.

Version: 046b26f

We intend to give a heuristic dictionary between statements in l-adic cohomology and statements in Hodge theory. This dictionary has, as its most notable sources sources, [3] and the conjectural theory of Grothendieck motives [2]. Up until now, it has mainly served to formulate conjectures in Hodge theory, and it has sometimes even suggested a proof.

## 1

A mixed Hodge structure H consists of

1. a \mathbb{Z}-module H_\mathbb{Z} of finite type (the “integer lattice”);

2. a finite increasing filtration W of H_\mathbb{Q}= H_\mathbb{Z}\otimes\mathbb{Q} (the “weight filtration”);

3. a finite decreasing filtration F of H_\mathbb{C}= H_\mathbb{Z}\otimes\mathbb{C} (the “Hodge filtration”).

This data is subject to the following condition: there exists a (unique) bigradation of \operatorname{Gr}_W(H_\mathbb{C}) by subspaces H^{p,q} such that

1. \operatorname{Gr}_W^n(H_\mathbb{C}) = \bigoplus_{p+q=n}H^{p,q}

2. the filtration F induces on \operatorname{Gr}_W(H_\mathbb{C}) the filtration \operatorname{Gr}_W(F)^p = \bigoplus_{p'\geqslant p} H^{p',q'}

3. \overline{H^{pq}}=H^{qp}.

A morphism f\colon H\to H' is a homomorphism f_\mathbb{Z}\colon H_\mathbb{Z}\to H'_\mathbb{Z} such that f_\mathbb{Q}\colon H_\mathbb{Q}\to H'_\mathbb{Q} and f_\mathbb{C}\colon H_\mathbb{C}\to H'_\mathbb{C} are compatible with the filtrations W and F (respectively).

The Hodge numbers of H are the integers h^{pq} = \dim H^{pq} = h^{qp}. \tag{1.2}

We say that H is pure of weight n if h^{pq}=0 for p+q\neq n (i.e. if \operatorname{Gr}_W^i(H)=0 for i\neq n). We also say that H is a Hodge structure of weight n.

The Tate Hodge structure \mathbb{Z}(1) is the Hodge structure of weight -2, purely of type (-1,-1), for which \mathbb{Z}(1)_\mathbb{C}=\mathbb{C} and \mathbb{Z}(1)_\mathbb{Z}= 2\pi i\mathbb{Z}= \operatorname{Ker}(\exp\colon\mathbb{C}\to\mathbb{C}^*)\subset\mathbb{C}. We set \mathbb{Z}(n)=\mathbb{Z}(1)^{\otimes n}.

We can show that mixed Hodge structures form an abelian category. If f\colon H\to H' is a morphism, then f_\mathbb{Q} and f_\mathbb{C} are strictly compatible with the filtrations W and F (cf. [1, s. 2.3.5]).

## 2

Let A be a normal integral ring of finite type over \mathbb{Z}, with field of fractions K, and \overline{K} an algebraic closure of K. Let K_{nr} be the largest sub-extension of \overline{K} that is unramified at each prime ideal of A. We know that, or we set, \pi_1(\operatorname{Spec}(A),\overline{K}) = \operatorname{Gal}(K_{nr}/K).

For every closed point x of \operatorname{Spec}(A), defined by some maximal ideal m_x of A, the residue field k_x=A/m_x is finite; the point x defines a conjugation class of “Frobenius substitutions” \varphi_x\in\pi_1(\operatorname{Spec}(A),\overline{K}). We set q_x=\#k_x and F_x=\varphi_x^{-1}.

Let K be a field of finite type over the prime field of characteristic p, let \overline{K} be an algebraic closure of K, let l be a prime number \neq p, and let H be a \mathbb{Z}_l- (or a \mathbb{Q}_l-) module of finite type endowed with a continuous action \rho of \operatorname{Gal}(\overline{K}/K). We will still suppose in what follows that there exists an A as above, with l invertible in A, and such that \rho factors through \pi_1(\operatorname{Spec}(A),\overline{K}) = \operatorname{Gal}(K_{nr}/K). We say that H is pure of weight n if, for every closed point x of an non-empty open subset of \operatorname{Spec}(A), the eigenvalues \alpha of F_x acting on H are algebraic integers whose complex conjugates are all of absolute value |\alpha|=q_x^{n/2}.

If the Galois module H “comes from algebraic geometry,” then there exists a (unique) increasing filtration W on H_{\mathbb{Q}_l}=H\otimes_{\mathbb{Z}_l}\mathbb{Q}_l (the “weight filtration”) that is Galois invariant and such that \operatorname{Gr}_n^W(H) is pure of weight n.

We can also further suppose that \operatorname{Gr}_n^W(H) is semi-simple.

When we have a resolution of singularities, we can often give a conjectural definition of W, whose validity follows from the Weil conjectures [5] (cf. §6).

Let \mu be the subgroup of \overline{K}^* given by the roots of unity. The Tate module \mathbb{Z}_l(1), defined by \mathbb{Z}_l(1) = \operatorname{Hom}(\mathbb{Q}_l/\mathbb{Z}_l,\mu) is pure of weight -2. We set \mathbb{Z}_l(n)=\mathbb{Z}_l(1)^\otimes n.

It is trivial that every morphism f\colon H\to H' is strictly compatible with the weight filtration.

Principle 2.1 agrees with the fact that every extension of \mathbb{G}_m (“weight -2”) by an abelian variety (“weight -1>-2”) is trivial.

## 3

The Galois modules that appear in l-adic cohomology have, as analogues, over \mathbb{C}, mixed Hodge structures. We further have the dictionary

 pure module of weight n Hodge structure of weight n weight filtration weight filtration Galois-compatible homomorphism morphism Tate module \mathbb{Z}_l(1) Tate Hodge structure \mathbb{Z}(1)

## 4

Let X be a complex algebraic variety (i.e. a scheme of finite type over \mathbb{C} that we assume to be separated). Then there exists a subfield K of \mathbb{C}, of finite type over \mathbb{Q}, such that X can be defined over K (i.e. it comes from an extension of scalars of K to \mathbb{C} applied to a K-scheme X'). Let \overline{K} be the algebraic closure of K in \mathbb{C}. The Galois group \operatorname{Gal}(\overline{K}/K) then acts on the l-adic cohomology groups H^\bullet(X,\mathbb{Z}_l); we have H^\bullet(X(\mathbb{C}),\mathbb{Z})\otimes\mathbb{Z}_l = H^\bullet(X,\mathbb{Z}_l) = H^\bullet(X'_{\overline{K}},\mathbb{Z}_l).

By §3, we should expect for the cohomology groups H^n(X(\mathbb{C}),\mathbb{Z}) to carry natural mixed Hodge structures. This is what we can prove (see [1, s. 3.2.5] for the case where X is smooth; the proof is algebraic, using classical Hodge theory [6]). For X projective and smooth, the Weil conjectures imply that H^n(X,\mathbb{Z}_l) is pure of weight n, while classical Hodge theory endows H^n(X,\mathbb{Z}) with a Hodge structure of weight n. For every morphism f\colon X\to Y, and for K large enough, f^\bullet\colon H^\bullet(Y,\mathbb{Z}_l)\to H^\bullet(X,\mathbb{Z}_l) Galois-commutes (by structure transport); similarly, f^\bullet\colon H^\bullet(Y,\mathbb{Z})\to H^\bullet(X,\mathbb{Z}) is a morphism of mixed Hodge structures. For X smooth, the cohomology class Z in H^{2n}(X,\mathbb{Z}_l(n)) of an algebraic cycle of codimension n defined over K is Galois invariant, i.e. it defines c(Z) \in \operatorname{Hom}_{\operatorname{Gal}}(\mathbb{Z}_l(-n),H^{2n}(X,\mathbb{Z}_l)). Similarly, the cohomology class c(Z)\in H^{2n}(X(\mathbb{C}),\mathbb{Z}) is purely of type (n,n), i.e. it corresponds to c(Z) \in \operatorname{Hom}_{\mathrm{H.M.}}(\mathbb{Z}(-n),H^{2n}(X(\mathbb{C}),\mathbb{Z})).

## 5

If f\colon H\to H' is a Galois-compatible morphism between \mathbb{Q}_l-vector spaces of different weights, then f=0. Similarly, if f\colon H\to H' is a morphism of pure mixed Hodge structures of different weights, then f is torsion. A more useful remark is

Let H and H' be Hodge structures of weight n and n' (respectively), with n>n'. Let f\colon H_\mathbb{Q}\to H'_\mathbb{Q} be a homomorphism such that f\colon H_\mathbb{C}\to H'_\mathbb{C} respects F. Then f=0.

## 6

Let X be a smooth projective variety over \mathbb{C}, let D=\sum_1^n D_i a normal crossing divisor in X, with D_i all smooth divisors, and let j be the inclusion of U=X\setminus D into X. For Q\subset[1,n], we set D_q=\bigcap_{i\in Q}D_i.

In l-adic cohomology, we canonically have R^q j_* \mathbb{Z}_l = \bigoplus_{\#Q=q} \mathbb{Z}_l(-q)_{D_Q} \tag{6.1} and the Leray spectral sequence for j is of the form E_2^{pq} = \bigoplus_{\#Q=q} H^p(D_Q,\mathbb{Q}_l)\otimes\mathbb{Z}_l(-q) \Rightarrow H^{p+q}(U,\mathbb{Q}_l). \tag{6.2}

By the Weil conjectures [5], H^p(D_Q,\mathbb{Q}_l) is pure of weight p, so that E_2^{pq} is pure of weight p+2q. As a quotient of a sub-object of E_2^{pq}, E_r^{pq} is also pure of weight p+2q. By §5, d_r=0 for r\geqslant 3, since the weights p+2q and p+2q-r+2 of E_r^{pq} and E_r^{p+q,q-r+1} (respectively) are different. Thus E_3^{pq}=E_\infty^{pq}. Up to renumbering, the weight filtration of H^\bullet(U,\mathbb{Q}_l) is the abutment of (6.2): \operatorname{Gr}_n^W(H^k(U,\mathbb{Q}_l)) = E_3^{2k-n,n-k}. \tag{6.3}

## 7

In integer cohomology, for the usual topology, the Leray spectral sequence for j is of the form 'E_2^{pq} = \bigoplus_{\#Q=q} H^p(D_Q,\mathbb{Z}) \Rightarrow H^{p+q}(U,\mathbb{Z}). \tag{7.1}

Since each D_Q is a non-singular projective variety, 'E_2^{pq} is endowed with a Hodge structure of weight p. We set E_2^{pq}='E_2^{pq}\otimes\mathbb{Z}(-q) (a Hodge structure of weight p+2q). As an abelian group, E_2^{pq}='E_2^{pq}; it is interesting to consider (7.1) as a spectral sequence with initial page E_2^{pq}. By §3, we should expect for d_2\colon E_2^{pq}\to E_2^{p+2,q-1} to be a morphism of Hodge structures. We prove this by thinking of d_2 as a Gysin morphism. Then E_3^{pq} is endowed with a Hodge structure of weight p+2q. By §3, we expect that, modulo torsion, the spectral sequence1 (6.2) degenerates at the E_3 page (i.e. E_3=E_\infty), and that the vanishing of the d_r (for r\geqslant 3) is an application of §5. This programme was successfully completed in [1, s. 3.2]. There, we define the weight filtration of H^\bullet(U,\mathbb{Q}) as the abutment of (7.1), up to renumbering (6.3).

In fact, to endow the cohomology groups H^\bullet with a mixed Hodge structure, the key point has always been, up until now, to find a spectral sequence E abutting to H^\bullet such that the l-adic analogue of E_2^{pq} be conjecturally pure (of weight p+2q); E_2^{pq} should then carry a natural Hodge structure (of weight p+2q), and the filtration W is the abutment of E.

## 8

Let \operatorname{Spec}(V) be the spectrum of a Henselian discrete valuation ring (a Henselian trait) with field of fractions K, and residue field k that is of finite type over the prime field of characteristic p. Let \overline{K} be an algebraic closure of K, and let H be a vector space of finite dimension over \mathbb{Q}_l (for l\neq p), on which \operatorname{Gal}(\overline{K}/K) acts continuously. By Grothendieck, we know ([4, Appendix]) that a subgroup of finite index of the inertia group I acts unipotently. By replacing V with a finite extension, we arrive to the case where the action of all of I is unipotent (the semi-stable case); it then factors as the largest pro-l-group I_l that is a quotient of I, and canonically isomorphic to \mathbb{Z}_l(1).

In the semi-stable case, if the Galois module H “comes from algebraic geometry,” then there exists a (unique) increasing filtration W of H (the “weight filtration”) such that I acts trivially on \operatorname{Gr}_n^W(H), and such that \operatorname{Gr}_n^W(H), as a Galois module under \operatorname{Gal}(\overline{k}/k)\simeq\operatorname{Gal}(\overline{K}/K)/I is pure of weight n.

We can compare this with Principle 2.1 and with the appendix of [4].

If we have a resolution of the singularities, then we can sometimes give a conjectural definition of W, whose validity follows from the Weil conjectures. With the help of the resolution and of Weil, it is sometimes easy to show that, in any case, H splits into pure Galois modules (under \operatorname{Gal}(\overline{k}/k)).

Suppose that H is semi-stable. For T\in I_t, we define \log T by the finite sum -\sum_{n>0}(\mathrm{Id}-T)^n/n. The map (T,x)\mapsto\log T(x) can be identified with a homomorphism M\colon \mathbb{Z}_l(1)\otimes H \to H. \tag{8.2} Since \mathbb{Z}_l(1) is of weight -2, we necessarily have (cf. §5) M(\mathbb{Z}_l(1)\otimes W_n(H)) \subset W_{n-2}(H), \tag{8.3} and M induces \operatorname{Gr}(M)\colon \mathbb{Z}_l(1)\otimes\operatorname{Gr}_n^W(H) \to \operatorname{Gr}_{n-2}^W(H). \tag{8.4}

If X is a non-singular projective variety over an algebraically closed field k_0, then we define L\colon \mathbb{Z}_l(-1)\otimes H^\bullet(X,\mathbb{Z}_l) \to H^\bullet(X,\mathbb{Z}_l) as being the cup product with the cohomology class with a hyperplane section. We note that there is a formal analogy between L and M; in the same way that M is defined by an action of \mathbb{Z}_l(1), we can think of L as being defined by an action of \mathbb{Z}_l(-1); L increases the degree by 2, and \operatorname{Gr}M (8.4) decreases it by 2.

## 9

Let D be the unit disc, D^*=D\setminus\{0\}, and X \begin{CD} X @>>> \mathbb{P}^r(\mathbb{C})\times D \\@VfVV @VV\mathrm{pr}_2V \\D @= D \end{CD} a family of projective varieties parameterised by D, with f proper, and f|D^* smooth.

Keeping the notation of §8, and recalling that, in the analogy between Henselian traits and small neighbourhoods of 0 in the complex line, we have the following dictionary (note that the spectrum of the ring of germs at 0 of holomorphic functions is a Henselian trait):

 D \operatorname{Spec}(V) D^* \operatorname{Spec}(K) a universal covering \widetilde{D^*} of D^* \operatorname{Spec}(\overline{K}) the fundamental group \pi_1(D^*) the inertia group I (with \pi_1(D^*)=\mathbb{Z}\simeq\mathbb{Z}(1)_\mathbb{Z}) (with I_l=\mathbb{Z}_l(1)) X a projective scheme X over \operatorname{Spec}(V) X^*=f^{-1}(D^*) X_K \widetilde{X}=X\times_D\widetilde{D^*} X_{\overline{K}} the local system R^if_*\mathbb{Z}|D^* the Galois module H^i(X_{\overline{K}},\mathbb{Z}_l H^i(\widetilde{X},\mathbb{Z}) H^i(X_{\overline{K}},\mathbb{Z}_l)

Note that \widetilde{X} is homotopically equivalent to each of the fibres X_t=f^{-1}(t) (for t\in D^*): H^i(X_{\overline{K}},\mathbb{Z}_l) is again analogous to H^i(X_t,\mathbb{Z}), and the transformation of the monodromy T corresponds to the action of I.

Here, again, we know that a subgroup of finite index of \pi_1(D^*) acts unipotently on H^i(\widetilde{X},\mathbb{Q})=H^i(X_t,\mathbb{Q}). We place ourselves in the semi-stable case, where all of \pi_1(D^*) acts unipotently (this reduces to replacing D by a finite covering), and let T be the action of the canonical generator of \pi_1(D^*).

By §3 and §8, we expect for H^i(\widetilde{X},\mathbb{Q})\simeq H^i(X_t,\mathbb{Q}) to be endowed with an increasing filtration W, for \operatorname{Gr}_n^W(H^i(\widetilde{X},\mathbb{Q})) to be endowed with a Hodge structure of weight n, for \log T(W_n)\subset W_{n-2}, and for \log T to induce a morphism of Hodge structures M_n\colon \mathbb{Z}(-1)\otimes\operatorname{Gr}_n^W(H^i) \to \operatorname{Gr}_{n-2}^W(H^i). We would further like for (8.2), and not just (8.3) and (8.4), to have an analogue.

We have in fact managed to define, for each vector u of the tangent space to D at \{0\}, a mixed Hodge structure \mathscr{H}_u on H^i(\widetilde{X},\mathbb{Z}). The filtration W and the Hodge structures on the \operatorname{Gr}_n^W(H^i) are independent of u, and the dependence on u of \mathscr{H}_u can be expressed simply in terms of T. Analogously to (8.2), we find that, for any u, \log T induces a homomorphism of mixed Hodge structures M\colon \mathbb{Z}(1)\otimes H^i(\widetilde{X},\mathbb{Z}) \to H^i(\widetilde{X},\mathbb{Z}).

Finally, the analogy in 8.5 is not misleading (but here, the fact that f|D^* is assumed to be proper and smooth is probably essential). We can prove that

(\log T)^k\colon \operatorname{Gr}_{n+k}^W(H^n(\widetilde{X},\mathbb{Q})) \to \operatorname{Gr}_{n-k}^W(H^n(\widetilde{X},\mathbb{Q})) is an isomorphism for all k (cf. [6, IV 6, Corollary to Theorem 5]). This characterises the filtration W. Up until the present, we have only had an analogue of the positivity theorem of Hodge (cf. [6, IV 7, Corollary to Theorem 7]) in very particular cases. We hope that the mixed structures \mathscr{H}_u determine the asymptotic behaviour, for t\to0, of the family of pure structures H^i(X,\mathbb{Z}) (for t\in D^*).

# Bibliography

[1]
P. Deligne. “Théorie de hodge.” Publ. Math. I.H.E.S. 40 (n.d.).
[2]
M. Demazure. “Motifs des variétés algébriques.” Sém. Bourbaki. (1969–70), Talk no. 365.
[3]
J.-P. Serre. “Analogues kählériens de certaines conjectures de weil.” Ann. Of Math. 71 (1960), 392–394.
[4]
J.-P. Serre, J. Tate. “Good reduction of abelian varieties.” Ann. Of Math. 88 (1968), 392–517.
[5]
A. Weil. “Number of solutions of equations in finite fields.” Bull. Amer. Math. Soc. 55 (1949), 497–508.
[6]
A. Weil. Introduction à l’étude des variétés kählériennes. Hermann, 1958. (1958).

1. [Trans.] The original refers to (6.4), but this seems to be a typo.↩︎