Translator’s note
This page is a translation into English of the following:
The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
Version: f36a214
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
a \mathbb{Z}-module H_\mathbb{Z} of finite type (the “integer lattice”);
a finite increasing filtration W of H_\mathbb{Q}= H_\mathbb{Z}\otimes\mathbb{Q} (the “weight filtration”);
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
\operatorname{Gr}_W^n(H_\mathbb{C}) = \bigoplus_{p+q=n}H^{p,q}
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'}
\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 sequence (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
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[4]
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[5]
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