Ω X generated locally in each adapted coordinate neighborhood as before by Ω and the forms

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1 Contents 1. Absolute Hodge cohomology of smooth complex varieties 1 2. Zariski descent 5 3. Multiplicative properties 7 4. Exercises 10 References Absolute Hodge cohomology of smooth complex varieties In this section we will show how to construct a complex that computes the real absolute Hodge cohomology of a smooth complex variety that is not necessarily proper. Let be a smooth complex variety of dimension d. By Nagata [Nag62] and Hironaka [Hir64] we can nd a proper complex variety and a dense open embedding j : such that D = \ is a simple normal crossing divisor (NCD). This means that, each point x has a coordinate neighbourhood U with coordinates (z 1,..., z d ) on which x has coordinates (0,..., 0), there is an integer k with 0 k d such that D U = {(z 1,..., z d ) z 1... z k = 0}, and each irreducible component D i of D is smooth (that is, the irreducible components of D do not have self intersections). Such coordinate neighborhood is called adapted to D. Denition 1.1. The sheaf of holomorphic forms with logarithmic poles along D, denoted Ω (log D) is the subsheaf of algebras of j Ω generated locally in each adapted coordinate neighborhood as before by Ω and the forms d z i z i, i = 1,..., k. The weight ltration W of Ω (log D) is the multiplicative ltration that assigns weight zero to the sections of Ω and weight one to the sections d z i/z i. The decalé ltration of W is denoted by Ŵ. The Hodge ltration F of Ω (log D) is the decreasing ltration F p Ω (log D) = Ω q (log D). q p Lemma 1.2. The inclusion Ω (log D) j Ω is a quasi-isomorphism. Therefore there are isomorphisms H (, Ω (log D)) H (, j Ω ) H dr(, C). Thus we can use dierential forms with logarithmic poles to compute de Rham cohomology of. The key point now is that this complex of forms allow us to relate the cohomology of to the cohomology of smooth proper complex varieties of dierent dimensions. To see this we dene the residue map. Let D = k i=1 D i be the decomposition of D into irreducible components. For each I {1,..., k} we denote by D I = i I D i 1

2 2 and we write D n = I =n where for a subset I as before we denote by I its cardinal. Then D n is a disjoint union of smooth proper complex varieties of dimension d n. Let U be a coordinate neighborhood with coordinates (z 1,..., z k ) adapted to D. For each j {1,..., k} we will denote by i j the index of the component of D satisfying D ij = {z j = 0}. If D I, η = α d z j 1 z j1 d z j m z jm then the residue is dened locally by W n Ω r (logd), m n Res(η) = { 0, if m < n, α Dij1 D ijn, if m = n. Lemma 1.3. The residue of a dierential forms with logarithmic poles denes an isomorphism Res: Gr W n Ω (log D) Ω Dn[ n]. Lemma 1.4. The natural maps are ltered quasi-isomorphisms. (Ω (log D), W ) (Ω (log D), τ) (j Ω, τ) Proof. The right arrow is a quasi-isomorphims because the inclusion Ω (log D) j Ω is a quasi-isomorphism. The right arrow is a ltered quasi-isomorphism because, by Lemma 1.3 { H r (Gr W n Ω (log D)) = 0, if r n, ι DnC Dn, if r = n. where ι Dn : D n is the map induced by the inclusion D and C Dn is the constant sheaf with ber C. Hence, the cohomology sheaf of Gr W n Ω (log D) is concentrated in degree n. Therefore H n (Gr W n and, for r n, Ω (log D)) = H n (Ω (log D)) = H n (Gr τ n Ω (log D)), H r (Gr W n Ω (log D)) = 0 = H r (Gr τ n Ω (log D)). We now choose asque resolutions of the sheaves A, A Q, C and Ω that t in a commutative diagram A A Q C Ω A A Q C Ω

3 3 of complexes of sheaves in. Consider the diagram (j C, τ) (j Ω, τ) (Ω (log D), τ) i 1 (Ω i 2 (log D), W ) The arrows in the above diagram are ltered quasi-isomorphisms and we will denote ( Ω, W ) = cone(i 1 i 2 ). From the diagram we obtain a ltered quasi-isomorphism (j C, τ) ( Ω, W ) and (Ω (log D), W ) ( Ω, W ). We now consider the diagram of complexes of sheaves on j (A Q ) ( Ω, W ) j (A ) (j (A Q ), τ) (Ω (log D), W, F ) that we denote D(, ). We choose compatible (ltered) asque resolutions on the sheaves of such diagram and denote the corresponding diagram by D(, ). We nally apply the functor of global sections Γ to the above complex to obtain a diagram of ltered complexes of A-modules that we denote by Γ(D(, ) ). Theorem 1.5. Let be a smooth complex variety and a compactication of with D = \ a NCD. The complex Γ(D(, ) ) is a mixed Hodge complex and its class in A HC does not depend on nor on the choices of asque resolutions. We now want to particularize to the case when A = R. In this case we can simplify the construction of a mixed R-Hodge complex by considering an acyclic resolution of the sheaf Ω (log D) that has a real structure. In this way we will be able to use the same complex to recover also the real piece of the cohomology, avoiding most of the technicalities of the previous construction. Denition 1.6. The sheaf of dierential forms with logarithmic singularities along D[BG94], denoted A (log D) is the subsheaf of algebras of j A generated locally in each adapted coordinate neighborhood as before by A and the forms d z i, d z i, log(z i z i ) i = 1,..., k. z i z i The weight ltration W of A (log D) is the multiplicative ltration that assigns weight zero to the sections of A and weight one to the sections d z i/z i, d z i / z i, and log(z i z i ). The decalé ltration of W is denoted by Ŵ. The Hodge ltration F of A (log D) is the decreasing ltration F p A (log D) = A q (log D). q p Clearly, A (log D) is stable under complex conjugation. We will denote by A (log D),R the subsheaf of algebras consisting of forms invariant under complex conjugation. Since the weight ltration is invariant under complex conjugation, it induces a ltration Ŵ on A (log D).,R

4 4 Since the complex A (log D) is a complex of A -algebras, it is a complex of ne sheaves and hence of acyclic sheaves. We will denote by A (log D) (respectively A (log D)) the complex of global,r sections of A (log D) (respectively A (log D)),R The main properties we need of the complex of dierential forms with logarithmic singularities are summarized in the following result. Theorem 1.7. With the hypothesis of Theorem 1.5 (1) The natural inclusion (Ω (log D), W, F ) (A (log D), W, F ) is a biltered quasi-isomorphism of complexes of sheaves. (2) The map A (log D),R S (, R) given by integration is a quasi-isomomorphims. Joining together theorems 1.5 and 1.7 we obtain Corollary 1.8. Let be a smooth complex variety and a compactication of with D = \ a NCD. Then the diagram ι (A (log D), Ŵ ) Id (A (log D), Ŵ ),R (A (log D), Ŵ, F ) is an R-Hodge complex that agrees with H(). The next task is to get rid of a particular compactication of to this end we consider the category C of compactications of. That is, the objects of C are triples (, D, ι) where is a proper complex variety, D is a NCD and ι: \ D is an isomorphism. A morphisms between (, D, ι) and (, D, ι ) is a morphism of varieties f : such that f ι = ι. The opposed category C is directed. Denition 1.9. The complex of dierential forms on with logarithmic singularities along innity is A log() = lim A (log D). C This complex inherits a weight ltration, a Hodge ltration and a complex conjugation, hence a real structure A log,r (). For any (, D, ι) C, the map (A log (), Ŵ, F ) (A (log D), Ŵ, F ) is a biltered quasi-isomorphism and the map (A log,r (), Ŵ ) (A (log D), Ŵ ) is a,r ltered quasi-isomorphism. Therefore the diagram (A log (), Ŵ ) A log () = ι Id (A log,r (), Ŵ ) (A log (), Ŵ, F ) is an R-Hodge complex that again agrees with H(). We can now apply Theorem 2.2 of the previous lecture to this complex.

5 5 Denition Let be a smooth complex variety. Then the absolute Hodge cohomology complex of is the complex of sheaves RAH C () is dened as RAH C () = Γ(D(A log ())). Unraveling this denition, we have RAH C () = cone(ŵ0a log,r () Ŵ0 F 0 A log () ϕ Ŵ0A log ())[ 1], where ϕ(r, f) = r f. More generally the twisted absolute Hodge cohomology complex of is In this case we obtain RAH C (, p) = Γ(D(A log ()(p))). RAH C (p)() = cone((2πi) p Ŵ 2p A log,r () Ŵ2p F p A log () ϕ Ŵ2pA log ())[ 1]. One of the main advantages of this construction is that it is functorial on the nose and we do not need to choose any particular asque resolution. Lemma Let Sm C be the category of smooth complex varieties. The assignment A log () is a presheaf of R-Hodge complexes in Sm C. The assignments RAH C (p)(), p Z are presheaves of complexes of real vector spaces. There is a variant of the absolute Hodge cohomology called weak Hodge cohomology or Deligne-Beilinson cohomology that is obtained by ignoring the eect of the weight ltration. The real Deligne-Beilinson cohomology of is H D(, R(p)) = H (cone((2πi) p A log,r () F p A log () ϕp A log ())[ 1])). We end this section discussing the case of real varieties. Let be a real variety. To it we can associate a complex variety C together with an conjugate-linear involution F of C. In fact the category of real varieties is equivalent to the category of complex varieties with a conjugate linear involution. On the space of dierential forms A ( C ) we dene an involution σ given by σ(ω) = F ω. that is, σ acts as complex conjugation on both, the space and the coecients of the dierential form. It is easy to see that σ induces an involution in the complexes RAH C (p)( C ). Denition Let be a smooth real variety. Then the absolute Hodge cohomology complex of is RAH(p)() = (RAH C (p)( C )) σ=id. That is, the complex formed by the elements of RAH C (p)( C ) that are invariant under σ. 2. Zariski descent One of the basic results in topology concerning singular cohomology is the Mayer- Vietoris sequence, that relates the cohomology of two open subsets with that of its union and intersection. This result is what makes singular cohomology easily computable. The Mayer-Vietoris theorem can be generalized as the ƒech spectral sequence in sheaf cohomology and in Deligne's theory of homological descent, that allow us to compute the cohomology of a variety in terms of hyper-resolutions. The fact that the ƒech spectral sequence associated to a Zariski open cover converges to the cohomology is a particular case of homological descent called Zariski descent.

6 6 We denote by Sm C the category of smooth complex varieties. Then RAH C (p) is a pre-sheaf on Sm C. That is, a contravariant functor. Let Ob(Sm C ) and let U = {U α } α Λ be a Zariski open covering of. The nerve of U, denoted N (U) is the simplicial smooth variety with N (U) n = U α0 U αn, (α 0,...,α n) Λ n the degeneracy map σ i : N (U) n N (U) n+1 sends the component U α0 U αn identically to the component U α0 U αi U αi U αn, while the face map δ i : N (U) n N (U) n+1 includes the component U α0 U αi U αn into the component U α0 Ûα i U αn, where the simbol means that the subset is omitted. If we apply the functor RAH C (p) to it we obtain a cosimplicial complex RAH C (p)(u). To this cosimplicial complex we can associate the total complex tot(rah C (p)(u)). Since RAH C (p) is a functor, there is a canonical map RAH C (p)() tot(rah C (p)(u)). Denition 2.1. Let F be a pre-sheaf of complexes on Sm C. satises Zariski descent if the canonical map is a quasi-isomorphims. F() F(U) Theorem 2.2. The pre-sheaf RAH C (p) satises Zariski descent. We says that F Proof. Since de Rham cohomology satises the Mayer-Vietoris theorem, the complexes A and A R satisfy descent for the ordinary topology, hence for the Zariski topology. Since A log () A () and A log,r () A R () are quasi-isomorphisms, then A log and A log,r satisfy Zariski descent. Since A log is a presheaf, to the simplicial variety N U we can associate a cosimplicial R-Hodge complex A log (N U). We now use the following result by Deligne. Theorem 2.3. Let H be a cosimplicial A-Hodge complex, F Q (F C, Ŵ ) H = ϕ 1 ψ 1 ϕ 2 (F Q, Ŵ ) F A Then the diagram tot(h) given by ψ 2 (F C, Ŵ, F ) ϕ 1 tot(f Q ) tot(f C, Ŵ ) ψ 1 ϕ 2 ψ 2 tot(f A ) tot(f Q, Ŵ ) tot(f C, Ŵ, F ) By this theorem, the total complex of A log (N U) is again an R-Hodge complex. There is a map of R-Hodge complexes H() tot A log (N U). This induces a map of mixed R-Hodge structures in cohomology. By Zariski descent of A R, this morphism induces an isomorphism of the corresponding real vector spaces. By Theorem 2.8 of the rst lecture, it is an isomorphism of mixed Hodge structures.

7 7 Therefore, for all p Z Ŵ2pA log,r, Ŵ 2p A log and Ŵ2p F p A log descent. Hence RAH C (p) satises Zariski descent. satisfy Zariski 3. Multiplicative properties Since absolute Hodge cohomology is the cohomology of a diagram of dierential graded commutative algebras it comes naturally with a graded commutative product that we will discuss in this section. Assume that we have a diagram F 1 F 2,... F k ϕ 1 ψ 1 ϕ 2 ψ 2 ψ k (3.1) D = F 0 F 1 F 2... F k such that all the entries F i and F i are dierential graded A-algebras and such that all the morphisms ϕ i and ψ i are morphisms of dierential graded A-algebras. Then the complex Γ(D) has several structures of graded algebras. Let α A and let ω = (ω 0,..., ω k, ω 1,..., ω k ) Γ n (D) and η = (η 0,..., η k, η 1,..., η k ) Γ m (D). Then we dene ω α η = (ω 0 η 0,..., ω k η k, ω 1((1 α)ϕ 1 (η 0 ) + αψ(η 1 )) + ( 1) n (αϕ 1 (ω 0 ) + (1 α)ψ 1 (ω 1 ))η 1,... ) The basic properties of this product are summarized in the following theorem. Lemma 3.1. (1) For every α, the map α is a morphism of complexes. (2) If α, α A then the morphisms α and α are homotopic. A homotopy is given, for ω and η as before, by h(ω α η) = (0,..., 0, (α α )ω 1η 1,..., (α α )ω kη k). Under the canonical automorphism T of Γ(D) Γ(D) given by T (ω η) = ( 1) deg ω deg η (η ω), the product α is transformed into 1 α. The product 0 and 1 are associative. As a direct consequence of this lemma we deduce that HAH(, A( )) = HAH(, n A(j)) n,j Z has a structure of bigraded associative algebra whose product is graded-commutative with respect to the rst degree. Nevertheless, at the level of complexes we do not have a product that is at the same time associative and graded commutative. For α = 0, 1 we have an associative product and if α = 1/2 A we have a graded commutative product, but none of the products α is at the same time associative and commutative. For the applications it will be useful to have a complex that computes real absolute Hodge cohomology and that is, at the same time, associative and graded commutative. Thus, we now will construct a pre-sheaf that is quasi-isomorphic to RAH C ( ) but that has a product that is at the same time associative and graded commutative. Let L denote the de Rham complex of algebraic dierential forms on A 1 R. That is L 0 = R[t] and L 1 = R[t] d t. Since Ŵ2pA log () and L are dierential graded

8 8 commutative algebras, the tensor product L Ŵ2pA log () (or rather the associated total complex) has a natural structure of dierential graded complex. Be aware of the usual sign convention for the tensor products. We will denote this product by. Denition 3.2. Let be a smooth complex variety. The complex IRAH C(p)() is the subcomplex of L Ŵ2pA log () composed by forms that satisfy ω t=0 (2πi) p Ŵ 2p A log,r (), ω t=1 Ŵ2p F p A log (). Clearly IRAH C(p)() is not just a subcomplex but also a subalgebra. Thus, it is a dierential graded commutative algebra. The elements of IRAH C(p)() are one parameter deformations between real forms and forms in a certain space of the Hodge ltration. Note that, instead of L we could consider dierential forms in the manifold R or just the interval [0, 1]. The important property of L is that it is a one-dimensional contractible complex. There are maps given by RAH C (p)() E I IRAH C (p)() E(r, f, η) = t f + (1 t) r + d t η, I(ω) = (ω t=0, ω t=1, t=1 t=0 where t=1 ( 1 (p(t) + q(t) d t) η = t=0 0 ω), ) q(t) d t η. Theorem 3.3. Let be a smooth complex variety. (1) The composition I E = Id RAH. (2) There is a homotopy h in IRAH such that E I Id IRAH = d h + h d. (3) If ω RAH n C(p)() and η RAH m C (q)(), then I(E(ω) E(η)) = ω 1/2 η. Proof. We check the rst statement. Let (r, f, η) RAH C (). Then E(r, f, η) = t f +(1 t) r+d t η. Since d t t=0 = d t t=1 = 0, we deduce that E(r, f, η) t=0 = r and E(r, f, η) t=1 = f. Since t=1 E(r, f, η) = η, the rst statement is clear. t=0 For each pair of integers n, p, we dene h: IRAH n C(p)() IRAH n 1 C (p)() by where t Then one can check easily that 0 h(ω) = t t=1 t=0 ω t 0 ω, ( t ) (p(t) + q(t) d t) η = q(s) d s η. h(d ω) + d h(ω) = E I(ω) ω. The third statement can also be checked easily. 0

9 9 This theorem has the following immediate consequence. Corollary 3.4. Let be a smooth complex variety. then H n (IRAH C (p)()) = H n AH(, R(p)). Moreover the product induced in HAH n (, R(p)) agrees with the product induced by Lemma 3.1. If is a smooth real variety, the involution σ determines an involution, also denoted σ, on IRAH C (p)( C ) that is compatible with the product. We dene IRAH(p)() = (IRAH C (p)( C )) σ=id. Corollary 3.5. Let be a smooth real variety. then H n (IRAH(p)()) = H n AH(, R(p)). Moreover the product of IRAH(p)() induces the usual product in HAH n (, R(p)).

10 10 4. Exercises Exercises (1) Let K be a number eld. Consider Q = Spec K as a variety over Q. Let R be the corresponding real variety. Compute the absolute absolute Hodge cohomology of R. Compare the obtained dimensions with the rank of the groups K i (K). (2) Compute the mixed Hodge structure of P 1 \ {0, 1}. References [BG94] J. I. Burgos Gil, A C -logarithmic Dolbeault complex, Compositio Math. 92 (1994), [Hir64] H. Hironaka, Resolution of singularities of an algebraic variety over a eld of characteristic zero, Annals of Math. 79 (1964), [Nag62] M. Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 110.