Hodge Theory I
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 (Serre 1960) and the conjectural theory of Grothendieck motives (Demazure 1969). Up until this point, 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'\geq 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, of pure degree (-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. [Hodge Theory II, Theorem 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_\mathrm{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_\mathrm{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-module (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_\mathrm{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 (Weil 1949) (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 $} 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 \operatorname{H}^\bullet(X,\mathbb{Z}_l); we have \operatorname{H}^\bullet(X(\mathbb{C}),\mathbb{Z})\otimes\mathbb{Z}_l = \operatorname{H}^\bullet(X,\mathbb{Z}_l) = \operatorname{H}^\bullet(X'_{\overline{K}},\mathbb{Z}_l).
By §3, we should expect for the cohomology groups \operatorname{H}^n(X(\mathbb{C}),\mathbb{Z}) to carry natural mixed Hodge structures. This is what we can prove (see [Hodge Theory II, Theorem 2.3.5] in the case where X is smooth; the proof is algebraic, using classical Hodge theory [W1958]). For X projective and smooth, the Weil conjectures imply that \operatorname{H}^n(X,\mathbb{Z}_l) is pure of weight n, while classical Hodge theory endows \operatorname{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 \operatorname{H}^\bullet(Y,\mathbb{Z}_l)\to \operatorname{H}^\bullet(X,\mathbb{Z}_l) Galois-commutes (by structure transport); similarly, f^\bullet\colon \operatorname{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 \operatorname{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),\operatorname{H}^{2n}(X,\mathbb{Z}_l)). Similarly, the cohomology class c(Z)\in \operatorname{H}^{2n}(X(\mathbb{C}),\mathbb{Z}) is of pure degree (n,n), i.e. it corresponds to c(Z) \in \operatorname{Hom}_{\mathrm{H.M.}}(\mathbb{Z}(-n),\operatorname{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 be 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} \operatorname{H}^p(D_Q,\mathbb{Q}_l)\otimes\mathbb{Z}_l(-q) \Rightarrow\operatorname{H}^{p+q}(U,\mathbb{Q}_l). \tag{6.2}
By the Weil conjectures (Weil 1949), \operatorname{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\geq3, 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 \operatorname{H}^\bullet(U,\mathbb{Q}_l) is the abutment of (6.2): \operatorname{Gr}_n^W(\operatorname{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} \operatorname{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, modulo torsion, for the spectral sequence (6.2)1 to degenerate at the E_3 page (i.e. E_3=E_\infty), and for the vanishing of the d_r (for r\geq3) to be an application of §5. This programme was successfully completed in [Hodge Theory II, §3.2]. There, we define the weight filtration of \operatorname{H}^\bullet(U,\mathbb{Q}) as the abutment of (7.1), up to renumbering (6.3).
In fact, to endow the cohomology groups \operatorname{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 then 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 ((Serre and Tate 1968, 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 (Serre and Tate 1968).
If we have a resolution of 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 \operatorname{H}^\bullet(X,\mathbb{Z}_l) \to \operatorname{H}^\bullet(X,\mathbb{Z}_l) as being the cup product with the cohomology class of 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 \operatorname{H}^i(X_{\overline{K}},\mathbb{Z}_l) |
| \operatorname{H}^i(\widetilde{X},\mathbb{Z}) | \operatorname{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^*): \operatorname{H}^i(X_{\overline{K}},\mathbb{Z}_l) is again analogous to \operatorname{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 \operatorname{H}^i(\widetilde{X},\mathbb{Q})=\operatorname{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 \operatorname{H}^i(\widetilde{X},\mathbb{Q})\simeq \operatorname{H}^i(X_t,\mathbb{Q}) to be endowed with an increasing filtration W, for \operatorname{Gr}_n^W(\operatorname{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(\operatorname{H}^i) \to \operatorname{Gr}_{n-2}^W(\operatorname{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 \operatorname{H}^i(\widetilde{X},\mathbb{Z}). The filtration W and the Hodge structures on the \operatorname{Gr}_n^W(\operatorname{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 \operatorname{H}^i(\widetilde{X},\mathbb{Z}) \to \operatorname{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(\operatorname{H}^n(\widetilde{X},\mathbb{Q})) \to \operatorname{Gr}_{n-k}^W(\operatorname{H}^n(\widetilde{X},\mathbb{Q})) is an isomorphism for all k (cf. (Weil 1958, IV 6, Corollary to Theorem 5)). This characterises the filtration W. Until now, we have only had an analogue of the positivity theorem of Hodge (cf. (Weil 1958, 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 \operatorname{H}^i(X,\mathbb{Z}) (for t\in D^*).
Bibliography
[Trans.] The original refers to (6.4) instead of (6.2), but this seems to be a typo.↩︎