ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES

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1 ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES MATT STEVENSON Abstract. This is the final project for Prof. Bhargav Bhatt s Math 613 in the Winter of 2015 at the University of Michigan. The purpose of this paper is to prove a result of Grothenieck [Gro66] showing, for a smooth scheme locally of finite type over C, the algebraic e Rham cohomology is isomorphic to the singular cohomology with C-coefficients) of its analytification. Our proof follows the exposition in [EC14, Chapter 2], an it is heavily-influence by [Gro66] an [Gil]. 1. Introuction On a smooth manifol X of real imension n, the sheaves k C,X of smooth k-forms on X give rise to the smooth e Rham complex C,X, where the ifferential of the complex is the exterior erivative. Taking the global sections of the smooth e Rham complex gives the complex of abelian groups ΓX, C,X ), an the k-th cohomology group Hk R X/R) := Hk ΓX, C,X)) is calle the e Rham cohomology of X. The classical e Rham theorem provies an isomorphism between the e Rham cohomology groups of X an the singular cohomology groups of X. Theorem 1. Let X be a smooth manifol, then H R X/R) Hsing X, R). This isomorphism associates a cohomology class [ω] HR k X/R) with the functional H kx, R) R given by [γ] ω. Moreover, the complex of sheaves C,X is a resolution of the sheaf R X of locally constant R-value functions on X: 0 R X 0 C,X γ 1 C,X... n C,X 0. The Poincaré lemma implies that the p C,X are acyclic the key here is that, in this context, there are partitions of unity). Therefore, the sheaf cohomology of X with coefficients in the constant sheaf R X can be compute from the cohomologies of the complex ΓX, C,X). This result is known as the e Rham theorem. Theorem 2. Classical e Rham Theorem) Let X be a smooth manifol, then H X, R X ) H R X/R). When one consiers instea a complex manifol X of complex) imension n, we still have an isomorphism between sheaf cohomology H X, C X ) of the constant sheaf C X an the singular cohomology H X, C) with C-coefficients. However, the rest of the story becomes more complicate. As before, the sheaves p holo,x of holomorphic p-forms on X give rise to the holomorphic e Rham complex holo,x, where the ifferential of the complex is the -operator. The holomorphic Poincaré lemma implies that the holomorphic e Rham complex is a resolution of the sheaf C X of locally constant C-value functions by coherent analytic sheaves: 0 C X 0 holo,x 1 holo,x... n holo,x 0. Date: April 24,

2 2 MATT STEVENSON In general 1, the sheaves p holo,x are not acyclic this correspons to the fact that there are no holomorphic partitions of unity on a complex manifol). Therefore, we cannot necessarily conclue that the sheaf cohomology groups H X, C X ) coincie with the cohomology groups of the complex ΓX, holo,x ) of global holomorphic forms. This problem isappears if we consier instea the hypercohomology H X, holo,x ) of the holomorphic e Rham complex, which oes inee calculate H X, C X ); this in turn is isomorphic to the singular cohomology group H X, C) with C-coefficients. Theorem 3. Analytic e Rham Theorem) Let X be a complex manifol, then H X, holo,x) H X, C). In [Ser56], Serre provie a ictionary linking the worl of complex manifols with that of smooth projective schemes over C. One may ask: what is the right notion of e Rham cohomology in the latter context, an o we get analogues of the e Rham theorems? As before, the right e Rham cohomology for a smooth scheme over C is the hypercohomology of the complex of sheaves of ifferentials, where now the ifferentials are algebraic instea of holomorphic. In fact, [Ser56] provies a natural map between this algebraic e Rham cohomology an the hypercohomology of the holomorphic e Rham complex on the associate analytic space. For a projective scheme, we show that this is an isomorphism this is our Theorem 7). The questions with which we are concerne in this paper is, when the smooth scheme over C is no longer assume to be projective, is there still an isomorphism between the algebraic e Rham cohomology an the singular cohomology of the analytification? The goal of this paper is to escribe the relationship between these two hypercohomologies, an ultimately to explain Grothenieck s proof from [Gro66] of this isomorphism. This will be an algebraic e Rham theorem, which we call Grothenieck s theorem. 2. Preliminaries 2.1. Complex Analytic Spaces. Let U C n be an open subset, an let O U be the sheaf of holomorphic functions on U. Take f 1,..., f m O U U), then the zero locus Z := {z U : f 1 z) =... = f m z) = 0} carries a natural structure sheaf O Z := O U /I Z, where I Z is the corresponing sheaf of ieals of Z. We say that Z, O Z ) is a complex analytic variety. Just as a scheme over C can be viewe as a locally ringe space that locally looks like a complex algebraic variety, a complex analytic space is a locally ringe space that locally looks like a complex analytic variety. These will be one of our main objects of stuy. The relationships between schemes over C an complex analytic spaces are etaile in e.g. [Ser56] or [sga71], but we will iscuss briefly some results that are neee for our purposes. Definition 1. A complex analytic space X, O X ) is a locally ringe space in which every point has an open neighbourhoo that is isomorphic as a locally ringe space) to a complex analytic variety. That is, for any x X, there exists an open neighbourhoo U of X so that U, O U ) is isomorphic to a complex analytic variety as locally ringe spaces. This isomorphism is calle a local moel at the point x. Often, the complex analytic space X, O X ) is enote just by X, omitting the structure sheaf. When referring to the sheaves of holomorphic ifferential forms on a complex analytic space X, we will enote it by X instea of holo,x when it is clear from context that X is analytic. 1 If X is a Stein manifol, then Cartan s theorem B [EC14, Theorem ] says that all coherent analytic sheaves on X are acyclic. In particular, the sheaves of holomorphic ifferentials holo,x are acyclic, so the analytic e Rham cohomology H X, holo,x ) coincies with H ΓX, holo,x )), which is analogous to the e Rham cohomology of the real case.

3 ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES 3 In the sequel, a scheme will refer to a scheme locally of finite type. Given a scheme X over C, there is a natural way to construct an associate analytic space, enote by X an, which is calle the analytification of X. We first construct the analytification when X is an affine scheme, an then procee to the general case. Consier the affine scheme X = Spec C[x 1,..., x n ]/f 1,..., f m ). The points of the analytification X an are the C-points of X, which is to say X an = XC) = {z C n : f 1 z) =... = f m z) = 0}. The structure sheaf O X an of X an is precisely the sheaf of holomorphic functions on XC), when viewe as an analytic subset of C n. Now, for an arbitrary scheme X, take any cover {U i } of X by open affine subsets an glue the locally ringe spaces {Ui an } to prouce the analytification X an of X. Example 1. Let X be an elliptic curve over C, i.e. the projective closure of an affine scheme of the form Spec C[x, y]/y 2 fx)) for some cubic polynomial fx) C[x]. The analytification of the elliptic curve X an = { [x: y : z] P 2 C): y 2 z = F x, z) }, where F x, z) is the homogenization of fx). This is a close Riemann surface of genus 1, hence iffeomorphic to a torus. Furthermore, there is a natural morphism φ: X an X of locally ringe spaces, which inuces a pullback functor on sheaves. For a sheaf F on X, we will enote the pull-back by F an := φ F = φ 1 F φ 1 O X O X an. Results relating a scheme an its analytification, as well those iscussing how sheaves behave uner this analytification proceure, are generally known as Serre s GAGA theorems, referring to [Ser56]. Two such results will be useful for our purposes. Theorem 4. Let X be a smooth projective scheme over C. The functor φ is an equivalence between the category of coherent sheaves of O X -moules on X an the category of coherent sheaves of O X an-moules on X an. Proof. See [Ser56, Theorem 2, Theorem 3] Even better, there is a relationship between the cohomology of the coherent sheaf an the cohomology of its pullback by the analytification functor. Theorem 5. Let X be a smooth projective scheme over C an let F be a coherent sheaf on X, then for any q 0, there is a natural isomorphism ɛ q : H q X, F) H q X an, F an ). Proof. See [Ser56, Theorem 1]. When X is a smooth projective scheme over C, the above theorem, along with the Hoge-to-e Rham egeneration, will suffice to show that the algebraic an the analytic e Rham cohomologies coincie. This is our Theorem Sheaves of ifferentials. Definition 2. Let A be a finitely-generate reuce C-algebra. The moule of Kähler ifferentials 1 A/C is the A-moule generate by the symbols {a: a A} moule an the following relations: 1) c = 0 for c C. 2) ab) = a b + b a for a, b A. There is a C-linear erivation : A 1 A/C with the universal property that any other C-linear erivation A M, where M is some A-moule, factors through : A 1 A/C in a unique way. Let i A/C := i 1 A/C, then the universal property shows that there are maps : i A/C i+1 A/C so that they form a complex A 1 A/C 2 A/C 3 A/C... 1)

4 4 MATT STEVENSON. Definition 3. On the affine scheme X = Spec A, the sheaf of Kähler ifferentials is X := A/C) 1 For a general scheme X, take any open affine cover an glue the sheaves of Kähler ifferentials on the open affines to prouce the sheaf 1 X. See [Har77, Section 2.8] for a more intrinsic escription of 1 X on a general scheme.) Moreover, the complex of moules Eq. 1) inuces a complex of sheaves of abelian groups, whose terms are coherent O X -moules 2 ) O X 1 X 2 X which is calle the algebraic e Rham complex of X. 3 X..., 2) There are some important properties of the sheaf of Kähler ifferentials on a smooth scheme X: first, 1 X is locally free of finite rank, hence coherent. Consequently, we can consier its analytification, along with the analytification of its exterior powers. In fact, the analytification of the sheaf q X/C = q 1 X/C of Kähler q-forms is the sheaf q X of holomorphic q-forms on the analytic space an Xan. See [Har77, Section 2.8] for a careful iscussion. In Section 3, we will efine the algebraic e Rham cohomology HR X/C) of a smooth scheme X over C to be the hypercohomology of the algebraic e Rham complex. When X is an affine scheme, this efinition reuces to the cohomology of the complex Eq. 1). In the following example, we see that the algebraic e Rham cohomology of an affine hyperelliptic curve coincies with the singular cohomology with C-coefficients of its analytification. Example 2. Let qx) C[x] be any monic polynomial of egree e with no multiple roots, an let A = C[x, y]/y 2 qx)). Then, X = Spec A is calle an affine hyperelliptic curve of egree e. The sheaf of ifferentials 1 X/C is a free A-moule of rank 1 an the algebraic e Rham cohomology is HR 0 X/C) C an H1 R X/C) Ce 1. A priori, the moule of Kähler ifferentials is 1 A/C = A x + A y 2yy q x)x). As qx) has no ouble roots, q x) an qx) are coprime, so there are polynomials ax), bx) C[x] so that 1 = ax)qx) + bx)q x). Let ω := ax)yx + 2bx)y, then we claim that 1 A/C = A ω. To see this, notice that x = ax)qx)x + bx)q x)x = ax)y 2 x + 2bx)yy = yω an y = ax)qx)y + bx)q x)y = ax)y 2 x + bx)q x)y = 1 2 ax)yq x)x + bx)q x)y = q x) 2 ω. In this case, the algebraic e Rham complex is just A 1 A/C = Aω. The kernel of is precisely the constants C A an the cokernel is the C-span of {ω, xω,..., x e 2 ω}. Therefore, the algebraic e Rham cohomology of X is { HRX/C) C, = 0 = C e 1, = 1. If we are instea concerne with smooth projective schemes over C, we coul compactify X to its projective closure in P 2 C) by aing 1 or 2 points epening on the egree e) an then use excision to compute the algebraic e Rham cohomology of the projective closure. Alternatively, one coul cover the projective closure by open affines compute the algebraic e Rham cohomology in a Čech-style.) Either way, we see that the algebraic e Rham cohomology of the projective closure also coincies with the singular cohomology of its analytification, which is a close Riemann surface of genus e 1 2 or e 2 2, epening on the parity of the egree e. 2 The algebraic e Rham complex is a complex of sheaves of abelian groups, where the morphisms are inuce by the erivation : A 1 A/C. Even though all of the terms of the complex are themselves coherent O X-moules, this is not necessarily a complex of coherent O X -moules. See [Gil, Example 1] for a counterexample.

5 ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES 5 3. Hypercohomology an Algebraic e Rham Cohomology Let ShvX) be the Grothenieck abelian category of sheaves of abelian groups on a locally ringe space X. Denote by ShvX) the category of cochain complexes over ShvX). Definition 4. For F ShvX), the k-th cohomology sheaf of the complex F is H k F ) := kerf k F k+1 )/imf k 1 F k ), viewe as quotient of sheaves. That is, H k F ) is the sheafification of the presheaf U kerf k U) F k+1 U)) imf k 1 U) F k U)). Moreover, a map of complexes F G is a quasi-isomorphism if the inuce map on cohomology sheaves H k F ) H k G ) is an isomorphsim for all k. Below we efine the hypercohomology of a complex in a rather non-constructive manner an refer the reaer to [Wei94, Section 5.7] for its usual escription as a collection of erive functors. Definition 5. The k-th hypercohomology is a functor H k X, ): ShvX) Ab satisfying the following two conitions: 1) If f : F G is a quasi-isomorphism of complexes, then H k X, f ) is an isomorphism. 2) If I is a complex of injectives, then H k X, I ) = H k ΓX, I )). As every complex of sheaves amits a quasi-isomorphism to a complex of injective sheaves, it follows from the efinition that the hypercohomology of any complex can be compute in terms of these injectives. Inee, if A ShvX) an A I is a quasi-isomorphism into some complex of injectives I, then H k A ) H k I ) = H k ΓX, I )). To compute hypercohomology, we can of course use the long exact sequence in cohomology associate to a short exact sequence of complexes. However, it will often be more helpful to buil a spectral sequence abutting to the esire cohomology group. There are 2 natural such spectral sequences, an at the moment we are concerne with just one of them. Proposition 1. For any F ShvX), there are convergent spectral sequences an E p,q 1 = H p X, F q ) H p+q X, F ) E p,q 2 = H p X, H q F )) H p+q X, F ). Proof. This is explaine for a general abelian category in [Wei94, Section 5.7]. Definition 6. Let X be the algebraic e Rham complex on a smooth scheme X over C. The algebraic e Rham cohomology HR X/C) is efine to be the hypercohomology of the algebraic e Rham complex. That is, HRX/C) k := H k X, X). In this context, the E 1 -spectral sequence of Proposition 1 is calle the Hoge-to-e Rham spectral sequence, an it takes the form E 1 p,q = H p X, q X ) Hp+q X, X). When the scheme X is sufficiently nice, this spectral sequence will egenerate at the E 1 -page i.e. the ifferentials r on the E r -page are zero for all r 1). Theorem 6. Hoge-to-e Rham egeneration) Let X be a smooth, proper scheme over C. Then, the Hoge-to-e Rham spectral sequence egenerates at the E 1 -page. Proof. See [Ill96, Theorem 6.9].

6 6 MATT STEVENSON In particular, when X is affine, the egeneration of the Hoge-to-e Rham spectral sequence implies that H k ΓX, X ) = Hk X, X ). Inee, the bottom row i.e. p = 0 row) on the E 1-page consists of the groups H 0 X, k ) an the ifferentials are horizontal. The egeneration at the E 1 -page means that the only terms on the E 2 -page are the groups H k ΓX, )) an the ifferentials are zero. It follows that these terms are isomorphic to those of the E -page, i.e. H k ΓX, )) H k X, ). Therefore, in the affine case, the algebraic e Rham cohomology HR X/C) can be thought of as very much the irect analogue of the usual e Rham cohomology, which is to say global close forms moule global exact forms. The egeneration of the Hoge-to-e Rham spectral sequence, combine with Theorem 5, is enough to show that the algebraic an analytic e Rham cohomologies coincie when X is also assume to be projective in fact, the projective assumption can be weakene to proper; see [sga71] for etails). Theorem 7. Let X be a smooth projective scheme over C, then H X, X ) H X an, Xan). In particular, the algebraic e Rham cohomology HR X/C) is isomorphic to the singular cohomology H X an, C) with C-coefficients. Proof. As in Proposition 1, there are the canonical E 1 -spectral sequences E p,q 1 = H p X, q X ) an E p,q 1,an = Hp X an, q X an) abutting to Hp+q X, X ) an Hp+q X an, Xan), respectively. Theorem 5 asserts that there are natural termwise-isomorphisms H p X, q X ) Hp X an, q Xan) for all p, q, as X is assume to be projective. This collection of isomorphisms yiels an isomorphism of spectral sequences {E p,q 1 } {E p,q 1,an }. In particular, there are isomorphisms of the abutments H X, X) H X an, Xan). 3) In Section 1, we remarke that the holomorphic Poincaré lemma implies that the augmentation map C X an X is a quasi-isomorphism, where an C X enotes the trivial complex 0 C an Xan 0. In particular, there are isomorphisms H X an, C) H X an, C X an) H X an, Xan). Composing these with Eq. 3) prouces the isomorphism H X an, C) HR X/C). 4. Grothenieck s Theorem Let X be a smooth scheme over C, then the analytification functor of Section 2 efines a map φ: X an X, which by Theorem 5 gives maps on the cohomologies H p X, q X ) Hp X an, q X an). Hence we get an inuce map on hypercohomology H X, X) H X an, Xan). 4) In [Gro66], Grothenieck asserte moreover that the morphism of Eq. 4) is an isomorphism, which shows that the algebraic e Rham cohomology an the analytic e Rham cohomology both capture the same information about the scheme X an its analytification X an. This theorem is our goal in this section; we follow the proof in [EC14, Chapter 2]. Theorem 8. Grothenieck) Let X be a smooth scheme over C. H X an, Xan) is an isomorphism. The natural map H X, X ) Recall that the holomorphic Poincaré lemma says that the augmentation map C X an Xan gives a quasi-isomorphism between the trivial complex C X : 0 C an X an 0 an the holomorphic e Rham complex Xan. Consequently, we get a chain of isomorphisms H X an, C X an) H X an, C X an) H X an, X an). Therefore, an immeiate corollary of Theorem 8 is that the algebraic e Rham cohomology HR X/C) coincies with the singular cohomology H X an, C) with C-coefficients. This is the algebraic e Rham theorem from Section 1.

7 ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES Reuction to the affine case. Let us assume that Theorem 8 hols for affine schemes, an use this to prove the general case. Let U = {U i } i I be an open affine cover of X, an for any list of inices i 0,..., i N I, let U i0,...,i N := U i0... U in. For a sheaf F of abelian groups on X an for each p 0, let Č p U, F) := FU i0,...,ip). i 0,...,i p I i 0<...<i p There are maps : Č p U, F) Čp+1 U, F) so that Č U, F) forms a complex of abelian groups calle the Čech complex of F these are the usual maps of the Čech complex; see [Har77, Chapter III.4] for the conventions). For each q 0, the above gives the Čech complex ČU, q X ) associate to q X ; these can be assemble into a ouble complex so that the ifferentials : q X q+1 X inuce maps Čp U, q X ) Čp U, q+1 X ) satisfying =. If K p,q := Čp U, p X ), then the ouble complex K, is of the form... Č 2 U, O X ) Č 2 U, 1 X ) Č 2 U, 2 X )... Č 1 U, O X ) Č 1 U, 1 X ) Č 1 U, 2 X )... Č 0 U, O X ) where all of the squares commute. Č 0 U, 1 X ) Č 0 U, 2 X ) Definition 7. The k-th Čech cohomology Ȟk U, X ) with respect to the open cover U) of the e Rham complex X is the k-th cohomology of the totalization TotK, ). Recall that the totalization of K, is the complex of abelian groups whose terms are TotK, ) k := Č p U, q X ) p+q=k an the ifferentials are given by + 1) p. We can also efine the Čech cohomology for any complex of sheaves of abelian groups in the same manner. See [EC14, Section 2.8] for further etails. Similar to how Čech cohomology of a quasi-coherent sheaf F computes the sheaf cohomology of F precisely because the cohomology of F on any piece of the affine cover is trivial, the Čech cohomology of the algebraic e Rham complex X will compute the hypercohomology because the sheaf cohomology of q X is trivial on any piece of the affine cover, for all q > 0. Theorem 9. For all k 0, there is an isomorphism Ȟk U, X ) Hk X, X ). Proof. See [EC14, Theorem 2.8.1]. Now, since Ȟ U, X ) is realize as the cohomology of the totalization of a ouble complex, there is a natural E 1 -spectral sequence converging to Ȟ U, X ), whose terms are given by E p,q 1 = H q Č p U, X) ) = q ΓU i0,...,i p, X) ). H i 0,...,i p I i 0<...<i p However, since U i0,...,i p is affine, there are by assumption isomorphisms H q ΓU i0,...,i p, X) ) H q Γ U i0,...,i p ) an, X an )). 5)...

8 8 MATT STEVENSON Why is this helpful? Well, the same argument can be repeate in the analytic case! The analytifications U an := {U i ) an } i I form an open cover of the analytic space X an by analytic varieties, an there are isomorphisms Ȟ U an, X an) H X an, Xan) between the Čech cohomology an the analytic e Rham cohomology. The E 1 -spectral sequence in this setting converges to Ȟ U an, Xan) an its terms are given by E p,q 1,an = Hq Č p U an, X an)) = q Γ U i0,...,i p ) an, )) X. an H i 0,...,i p I i 0<...<i p By our previous consierations, the two spectral sequences {E p,q 1 } an {Ep,q 1,an } are isomorphic because of the termwise isomorphisms of Eq. 5) an the ifferentials clearly commute). In particular, their E -pages are isomorphic, so for any k 0, H k X, X) Ȟk U, X) Ȟk U an, X an) Hk X an, X an) There is a slight problem with this reuction as state, namely that the intersection of the open affines U i0,...,i N s are not necessarily themselves affine because we have not mae the assumption that X is separate. To get aroun this, we shoul first repeat the above argument for separate schemes X, an then the general case follows by repeating the argument once more since the U i0,...,i N s will certainly be separate, if not affine. Therefore, we have shown that it suffices to prove Grothenieck s theorem our Theorem 8) for a smooth affine scheme over C. This is one in the subsequent subsection The affine case. In the previous subsection, we reuce the proof of Theorem 8 to the case of an affine scheme. In this setting, the analytic e Rham theorem allows us to reformulate Theorem 8 to the folllowing statement. Theorem 10. Let X be a smooth affine scheme over C, then for any k 0, there are isomorphisms H k X, X) H k X an, C). 6) It will be helpful to realize X as living insie some smooth projective scheme; to o so, we will nee to apply a form of) Hironaka s theorem on resolution of singularities. Hironaka s theorem asserts that there exists a smooth projective scheme Y over C an an open embeing i: X Y so that the complement U := Y X, a close subset of Y, is a normal crossing ivisor in Y. See [Hir64] for the proof of this result, an see [Har77, Chapter V.3, Remark 3.8.1] for a iscussion of normal crossing ivisors. Analytifying all of the above spaces, there is an embeing i: X an Y an. Originally, we were concerne with holomorphic forms on X an, but it is avantageous to push forwar the problem to Y an by consiering a sheaf of forms on Y an instea. With this in min, we efine k Y an U an ) to be the subsheaf of meromorphic k-forms on the complex analytic space Y an which restrict to holomorphic k-forms on X an. Sai ifferently, k Y an U an ) is the subsheaf of meromorphic k-forms on Y an that can only have poles on the complement U an = Y an X an of X an. In orer to say something intelligent about the hypercohomology of the complex Y an U an ), we can realize the terms k Y an U an ) as subsheaves of ostensibly simpler sheaves in such a way that the inclusion of complexes is a quasiisomorphism. Let k C,X an be the sheaf of smooth C-value k-forms on the analytic space Xan, then k Y an U an ) is clearly a subsheaf of i k C,X an ), since sections of k Y an U an ) are holomorphic in particular, smooth) k-forms when restricte to X an. The inclusions k Y an U an ) i k C,X an ) of sheaves on Y an inuce an inclusion of complexes Y an U an ) i C,X an ) of sheaves on Y an. Theorem 11. Atiyah-Hoge) The inclusion Y an U an ) i C,X an ) is a quasi-isomorphism.

9 ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES 9 Proof. See [HA55, Section 3]. Theorem 11 is sometimes calle the Funamental Lemma of Atiyah & Hoge, or the analytic) Atiyah-Hoge Lemma. The upshot of Theorem 11 is that there is an isomorphism in hypercohomology H Y an, Y an U an )) H Y an, i C,X an ) ). 7) Our goal is to first construct an isomorphism ) H Y an, Y an U an )) H ΓX, X )) an then buil an isomorphism H Y an, i C,X ) H X an, C). These combine with Eq. 7) to prove Theorem 10. an For a non-negative integer n, let k Y nu) an k Y annu an ) be the subsheaves of k Y U) an k Y an U an ) with poles of orer less than or equal to n on U an U an, respectively. It follows that we can write k Y U) = lim n k Y nu) an k Y an U an ) = lim k n Y annu an ). This escriptions of Y U) an Y an U an ) as filtere colimits of coherent sheaves is a crucial ingreient in the proof of the next two propositions. Proposition 2. There are isomorphisms H Y, Y U)) H Y an, Y an U an )). Proof. As Y is smooth an projective, Theorem 5 says that H p Y, q Y nu)) Hp Y an, q Y annu an )) for any n 0. Moreover, cohomology commutes with filtere colimits at least in this case, since Y an is compact an Y is a noetherian space), there are isomorphisms H p Y, q Y U)) = Hp Y, lim q n Y nu)) = lim H p Y, q Y nu)) n lim n H p Y an, q Y annu an )) = H p Y an, lim n q Y annu an )) = H p Y an, q Y an U an )). However, H p Y, q Y U)) an Hp Y an, q Y an U an )) are the p, q)-terms of the E 1 -page of the first quarant spectral sequences associate to an converging to) the hypercohomologies H p+q Y, Y U)) an H p+q Y an, Y an U an )), respectively. As before, the termwise isomorphism of the spectral sequences gives an isomorphism on the E -pages, i.e we have the esire isomorphisms H Y, Y U)) H Y an, Y an U an )). Proposition 3. There are isomorphisms H Y, Y U)) H X, X ). Proof. The plan is to buil the Čech complexes for H Y, Y U)) an H X, X ), respectively, as in Definition 7 an show that all of the terms of the two complexes are isomorphic. To ensure that the cohomology of the totalization of the Čech ouble complex oes yiel the correct hypercohomology, we nee to first check that the sheaf Y U) is acyclic on an affine open subset of Y we alreay checke this for X on X). Inee, let V Y be an affine open, then applying the fact that cohomology commutes with filtere colimits once again we fin that for all n, q H p V, q Y U)) = Hp V, lim q n Y nu)) lim H p V, q Y nu)) = 0, n where H p V, q Y nu)) = 0 because q Y nu) is coherent. Therefore, we can take any open affine cover U = {U i } i I of Y an the Čech cohomology with respect to U computes the hypercohomology, i.e. H Y, Y U)) Ȟ U, Y U)). As the pushforwar i X coincies with Y D), the terms of the Čech ouble complex associate to Y U) with respect to U can be written as Č p U, q Y U)) = Čp U, i q X ) = ΓU X, q i0,...,ip X ). 8) i 0,...,i p I i 0<...<i p

10 10 MATT STEVENSON Furthermore, let U X := {U i X} i I be the restriction of the cover U to X, then the terms of the Čech ouble complex associate to X with respect to U X can be written as Č p U X, q X ) = q ΓU i0,...,ip X, X ). 9) i 0,...,i p I i 0<...<i p However, the terms of the Čech ouble complexes Eq. 8) an Eq. 9) are ientical, so there is an equality between the Čech cohomologies, i.e. Ȟ U, Y U)) = Ȟ U X, X ). In particular, there is the esire isomorphism H X, X) Ȟ U X, X) = Ȟ U, Y U)) H Y, Y U)). Theorem 11, Proposition 3, an Proposition 2 combine to give the string of isomorphisms H X, X) H Y, Y U)) H Y an, Y an) H Y an, i C,X an ) ). Therefore, to prove Theorem 10, it suffices to show that H Y an, i C,X an) H X an, C), where the latter term is the singular cohomology of the analytic space X an with C-coefficients. This is precisely the assertion of our final proposition. Proposition 4. There are isomorphisms H Y an, i C,X an ) ) H X an, C). Proof. Pointwise multiplication gives a natural O C,Y an-moule structure to the sheaf i ) C,X, ) an which inicates that i C,X is acyclic inee, the an OC,Y an-moule structure allows us to use partitions of unity). Consequently, the hypercohomology is nothing more than H k Y an, i C,X an) Hk ΓY an, i C,X an ) )) H k ΓX an, C,X an)) Hk X an, C), where the last isomorphism is the classical e Rham theorem. References [EC14] P. A. Griffiths L. D. Trang E. Cattani, F. El Zein, eitor. Hoge Theory. Princeton University Press, [Gil] W. D. Gillam. On the e rham cohomology of algebraic varieties. Lecture Notes. [Gro66] A. Grothenieck. On the e Rham cohomology of algebraic varieties. Inst. Hautes Étues Sci. Publ. Math., 291):95 103, [HA55] W. V. D. Hoge an M. F. Atiyah. Integrals of the secon kin on an algebraic variety. Ann. of Math. 2), 62:56 91, [Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heielberg, Grauate Texts in Mathematics, No. 52. [Hir64] Heisuke Hironaka. Resolution of singularities of an algebraic variety over a fiel of characteristic zero. I, II. Ann. of Math. 2) ), ; ibi. 2), 79: , [Ill96] Luc Illusie. Frobenius et égénérescence e Hoge. In Introuction à la théorie e Hoge, volume 3 of Panor. Synthèses, pages Soc. Math. France, Paris, [Ser56] Jean-Pierre Serre. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble, 6:1 42, [sga71] Revêtements étales et groupe fonamental. Springer-Verlag, Berlin-New York, Séminaire e Géométrie Algébrique u Bois Marie SGA 1), Dirigé par Alexanre Grothenieck. Augmenté e eux exposés e M. Raynau, Lecture Notes in Mathematics, Vol [Wei94] Charles A. Weibel. An introuction to homological algebra, volume 38 of Cambrige Stuies in Avance Mathematics. Cambrige University Press, Cambrige, 1994.