THE NOETHER-LEFSCHETZ THEOREM

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1 THE NOETHER-LEFSCHETZ THEOREM JACK HUIZENGA Contents 1. Intoduction 1 2. Residue and tube ove cycle maps 2 3. Hodge stuctues on the complement 8 4. Peiod maps and thei deivatives Tautological defomations of hypesufaces The main theoem 13 Appendix A. Spectal sequences 16 A.1. Fundamental Notions 16 A.2. Hypecohomology 24 A.3. Abstact de Rham theoems 26 A.4. The spectal sequence of a filteed complex of sheaves 28 Refeences Intoduction In this pape, we collect and flesh out esults fom Calson et al. [1] to give a elatively self-contained poof of the Noethe-Lefschetz theoem via Hodge theoy and defomation theoy. The pincipal esult is the following theoem. Theoem (Noethe-Lefschetz). Assume that d 4. Outside a countable union of pope subvaieties in the paamete space of degee d sufaces in P 3, the Picad goup of the coesponding suface X is isomophic to Z, geneated by O X (1). Hence evey cuve on X is a complete intesection. The fist six sections, and fist half of this pape, compise the esults fundamentally elated to the poof of the theoem, culminating with the poof in Section 6. Date: Apil 6,

2 2 JACK HUIZENGA The second half of the pape is an appendix on the theoy of spectal sequences, in which we develop the algebaic techniques necessay fo the poof of the theoem. The appendix culminates with a theoem that allows the constuction of a spectal sequence of abelian goups associated to a filteed complex of sheaves. The Hodge to de Rham spectal sequence is seen to be a special case of this geneal constuction. Anothe example of this spectal sequence is cucial to ou poof of the Noethe-Lefschetz theoem. 2. Residue and tube ove cycle maps Ou goal in this section is to ty and undestand the cohomology of X in tems of the cohomology of the complement U = P n+1 \ X. Thee ae a couple of tools fo doing this. The most fundamental is the esidue map, which takes a p-fom on P n+1 with at most a simple pole along X and gives a holomophic (p 1)-fom on X. The othe tool is the tube ove cycle map τ : H k 2 (X; Q) H k 1 (U; Q). This map acts by taking a homology class in X and taking a sufficiently small cicle bundle ove the homology class, viewed as a subset of the bounday of the disk bundle ove X in P n+1. Conside the long exact sequence in cohomology of the pai (P n+1, U). This is by definition the long exact sequence H k 1 (U) H k (P n+1, U) H k (P n+1 ) H k (U). Using the Thom isomophism theoem and excision, we can eplace H k (P n+1, U) with H k 2 (X). When we do so, the esulting long exact sequence is H k 1 (U) H k 2 (X) i H k (P n+1 ) j H k (U). Hee is the adjoint of the afoementioned tube ove cycle map τ, i is the Gysin map in cohomology associated to the inclusion i : X P n+1, and j is the induced map in cohomology of the inclusion j : U P. The tube ove cycle map and its tanspose helps us to compute the pimitive middle cohomology H n va(x) of X. Poposition 2.1. The map : H n+1 (U) H n (X) is injective, with image H n pim(x). In shot, : H n+1 (U) H n pim(x) is an isomophism. Moeove, the cohomology H n (X) splits as H n (X) = H n pim(x) H n fix(x), whee H n fix (X) = i H n (P n+1 ). This splitting is othogonal.

3 Poof. We examine the Gysin sequence THE NOETHER-LEFSCHETZ THEOREM 3 H n+1 (P n+1 ) H n+1 (U) H n (X) H n+2 (P n+1 ). Fist suppose n is odd. Then H n+2 (P n+1 ) vanishes and so is sujective. Its image is Hpim(X), n since all the cohomology of X is pimitive. Fo injectivity, it is equivalent to see that the map H n+1 (P n+1 ) H n+1 (U) is zeo. Then we can instead show H n 1 (X) H n+1 (P ) is sujective. This map is Poincaé dual to the isomophism H n 1 (X) H n 1 (P n+1 ), so is in fact sujective. If instead n is even, then is injective. Its image is the kenel of i. So we need to show that the kenel of i is the same thing as pimitive cohomology. Suppose that α H n (X) is a cohomology class, with Poincaé dual γ. Wite ω H 2 (P n+1 ) fo the cohomology class of a hypeplane H tansvese to γ such that the hypeplane section H X is also tansvese to γ in X. Then i ω is the cohomology class of the hypeplane section H X. The Gysin homomophism i : H n+2 (X) H n+4 (P n+1 ) is Poincaé dual to the isomophism i : H n 2 (X) H n 2 (P n+1 ), so i (α i ω) = 0 if and only if α i ω = 0. Now we claim that i (α i ω) = i (α) ω. Indeed, i (α i ω) = [i [α i ω]] = [i (γ (H X))] = [i γ i (H X)] = [i γ H] = i α ω, whee backets denote Poincaé dual. But i α ω = 0 if and only if i α = 0, so α i ω = 0 if and only if i α = 0. Thus the kenel of i is the pimitive cohomology Hpim(X). n Fo the decomposition theoem, we obseve that X has no pimitive cohomology in dimensions othe than n and 0. It follows fom the Lefschetz decomposition theoem that H n (X) = Hpim(X) n Hfix n (X). In fact, the decomposition is othogonal, fo if α is pimitive and β = i ω n/2 then i (α β) = i α ω n/2 = 0 since α is in the kenel of the Gysin homomophism. Fo esidue maps, the setup is the following. Suppose that α is a diffeential p-fom on P n+1 with at most a logaithmic pole along X. If V P n+1 is a coodinate open set whee X has equation f = 0 and thee ae local coodinates (x 0 = f, x 1,..., x n ), we can uniquely wite fα = df η + η

4 4 JACK HUIZENGA whee η and η ae holomophic foms involving only dx 1,..., dx n. Then α = df f η + η f. When we take the exteio deivative of this expession, the fist summand still has a pole of ode at most 1 along X since wheneve 1/f appeas df does as well, so that no patial deivatives of 1/f ae taken with espect to f. But we will get a second ode pole along X fom the second tem unless if all the coefficient functions of η ae divisible by f. So in ode fo fdα to be holomophic, it is necessay that η /f be holomophic. Thus any α with at most a log-pole along X can be witten uniquely in the fom α = df f η + η fo holomophic (p 1)-foms η and η involving only x 1,..., x n. We then let (Res α) V X = η V X, and patch togethe the diffeent expessions ove an open coveing of P n+1 to fom Res α. Lemma 2.2. Res α is a well-defined holomophic (p 1)-fom on X. Poof. Notice that η and η ae in fact uniquely detemined by f, α, V, and the choice of coodinate system, since fα is a holomophic p-fom. Fist we show that (Res α) V X is independent of the choice of defining equation f. If f also defines X on V, we also have coodinates (f, x 1,..., x n ) on V ; thee is no need to change the othe n coodinates. Now thee is a nowhee vanishing function u such that f = uf. Then df f = d(uf ) uf = 1 uf (u df + du f ) = df f + du u. If η and η coespond to the choice of equation f, then it follows that α = df η + du ( df f u η + η = + 1 ) du f u df df η +, whee the tems in the involve only dx 1,..., dx n. We can ewite this as df (1 + f ) du η + = df (1 + f ) du η +, f u df f u df so the η fo f is (1 + f u 1 (du/df ))η. As f = 0 on X, the two η s agee when esticted to X. Next we show that Res α is independent of the choice of the coodinates (x 1,..., x n ), having aleady established independence of the choice of x 0 = f. So suppose we choose

5 THE NOETHER-LEFSCHETZ THEOREM 5 the system diffeently as y 1,..., y n. We can wite α = df f ψ + ψ, whee ψ and ψ involve only dy 1,..., dy n. We ae to show that ψ X = η X. Then dy i = y i f df + j y i x j dx j = j y i x j dx j since f is a coodinate in both coodinate systems. So we see that ψ and ψ aleady only involve the dx j. This foces ψ = η and ψ = η by the uniqueness emak in the fist sentence of the poof. Since the constuction is independent of the coodinate system, it follows that the foms glue togethe on ovelaps to give a globally defined holomophic (p 1)-fom on X. The fundamental esult elating the tube ove cycle and esidue maps is the Residue Fomula. Theoem 2.3 (Residue Fomula). Let α Ω p P (X) be a holomophic p-fom on P with at most a simple pole along X. Fo any homology class γ H p 1 (X), γ Res α = 1 2πi Poof. Fist we must give a moe detailed desciption of what we mean by τ(γ). This was defined to be the homology class of the bounday of a sufficiently small tubula neighbohood of γ in the estiction of the nomal bundle of X in P to γ. Hee, sufficiently small means that we make the neighbohood smalle and smalle until the homology class emains constant as we make it smalle. Assume that γ is a compact submanifold of X. Cove X by finitely many coodinate open sets U i whee X is given by f = 0. Let Ũi be open subsets of P such that Ũ i is an open neighbohood of U i in P and Ũi lies ove U i unde the identification of a neighbohood of the zeo section of N X/P with a neighbohood of X. Let ξ i be a patition of unity subodinate to {U i } on X, and let ξ i be the pullback to Ũi unde the pojection Ũi U i. If τ(γ) α = df f η + η, α. then ξ i α = df f ξ i η + ξ i η,

6 6 JACK HUIZENGA which poves that Res ξ i α = ξ i Res α. The upshot is that it suffices to show in local coodinates (f, x 1,..., x n ) that if α is compactly suppoted on the hypeplane f = 0, then γ 1 Res α = lim α, ε 0 2πi τ ε (γ) whee by τ ε (γ) we mean the set of points ove γ with f = ε. Since we don t need to use holomophic coodinates (we e aleady using a patition of unity, so they will be useless), we may in fact assume that γ is specified by the vanishing of some of the coodinate functions x i. The disk bundle ove γ is given by points ove γ with f < ε. The integal of η ove τ(γ) vanishes because a vecto in the span of / f and / f is in the tangent space to τ(γ) at each point, so that η esticts to 0 on τ(γ). As ε becomes small, the coefficients of η becomes essentially independent of f, since they must convege to the coefficients of η X. Viewing η X locally as a fom on P by pullback fom the pojection onto X, we ae theefoe justified in assuming the coefficients of η ae independent of f. But then τ(γ) α = τ(γ) df f η = df η = 2πi Res α S f 1 γ γ by Fubini s theoem. Simila to the Gysin sequence is the long exact sequence in hypecohomology associated to the shot exact sequence of chain complexes of sheaves 0 Ω P n+1 Ω Res Pn+1(log X) Ω 1 X 0. This is in fact a shot exact sequence, fo the kenel of Res locally consists of foms α = (df/f) η +η whee η vanishes along X, so that all the coefficients of η ae divisible by f and α is holomophic. The tems in the long exact sequence of hypecohomology ae the same as the tems of the Gysin sequence, due to the following esult.

7 THE NOETHER-LEFSCHETZ THEOREM 7 Theoem 2.4. (1) Thee is a commutative diagam H k+1 (U) H k i (X) H k+2 (P n+1 ) j H k+2 (U) 1 2πi H k+1 (U; C) Res H k (X; C) δ H k+2 (P n+1 ; C) j H k+2 (U; C) H k+1 (Ω P n+1(log X)) Res H k (Ω X) (2) Thee ae natual isomophisms δ H k+2 (Ω P n+1) H p (Ω P n+1(log X)) H p (Ω P n+1( X)) H p (Ω U) H p (U; C). H k+2 (Ω Pn+1(log X)). In fact, all fou of the vetical maps in the bottom ow of the above diagam ae isomophisms. Poof. (1) Commutativity of the uppe left squae follows fom Theoem 2.3 since this fomula shows that Res is the tanspose of τ up to a facto of 2πi. The connecting mophism in the middle is defined by the connecting mophism on the bottom, but moally should be the same as the connecting mophism in the exact sequence of the pai (P, U), up to the Thom isomophism. The top i was also defined by this popety, although it happens to be the Gysin homomophism. The lowe ight squae is the othe slightly nontivial one, but the top map is defined by estiction of foms, while the bottom is just an inclusion. It petty clealy commutes. The vetical maps that ae noted to be isomophisms ae in fact isomophisms by the Abstact de Rham Theoem A.3.1, as shown in the appendix on spectal sequences. (2) We fist note that H p (Ω U ) is isomophic to Hp (U; C) whethe we view Ω U as a sheaf on P n+1 o as a sheaf on U. This is because Ω U (V ) = Ω U (V \X) fo any open V Pn+1, so that if we fix a good cove of P n+1 whose intesections with U ae simultaneously good fo U, then all the tems in the Čech-de Rham double complex fo Ω U ae equal, while it follows fom the abstact de Rham theoem that H p (U, Ω U ) = H p (U; C). By applying the five-lemma to the diagam of the pevious pat, we see that the composition H p (Ω P (log X)) H p (U; C) is an isomophism. Also the final map n+1 H p (Ω U ) Hp (U; C) is an isomophism. To pove that the othe two maps ae isomophisms, it is good enough to pove that one of them is an isomophism. So we will pove that thee is an isomophism H p (Ω P (log X)) H p (Ω n+1 P ( X)). n+1

8 8 JACK HUIZENGA Fist give Ω P n+1 ( X) the ode of pole filtation F p pole (Ω P n+1( X)) = (0 0 0 Ωp P n+1 (X) Ω p+1 P n+1 (2X) ) and give Ω P n+1 (log X) the tivial filtation. The inclusion Ω P n+1 (log X) Ω P n+1 ( X) induces a filteed quasi-isomophism. That is, the sequence 0 Ω p P n+1 (log X) Ω p P n+1 (X) Ω p+1 P n+1 (2X)/Ω p+1 P n+1 (X) is exact. Exactness at the fist two stages is clea. Fo exactness at the highe stages, we wok in local coodinates. Let ω Ω p+k P ((k+1)x), and suppose dω Ω p+k+1 n+1 P ((k+1)x). n+1 Choose ou coodinate system to be (x 0 = f, x 1,..., x n ). Then we can wite ω = g I dx I I =p+k Modulo Ω p+k P (kx), we can assume that the g n+1 I s ae independent of x 0 fo all I, and it still follows that dw Ω p+k+1 P ((k + 1)X). When we explicitly diffeentiate ω, the n+1 only tems with poles of ode geate than k + 1 come fom the patial deivative with espect to x 0. So the condition that dω = 0 in the quotient is exactly the condition that g I is divisible by x 0 if I does not contain 0. But g I is independent of x 0, so then must be zeo. Theefoe evey tem of ω contains dx 0, and we can wite ω = dx 0 ψ fo some x k+1 0 holomophic fom ψ with no dx 0 s and whose coefficient functions ae independent of x 0. Now conside the fom 1 k x k 0 ψ. Clealy ( d 1 ) k x k 0 ψ = dx 0 ψ 1 x k+1 0 k x k 0 dψ. Since ψ is holomophic, we conclude that d( 1 k x k 0 ψ) = ω mod Ω p+k P (kx), the sequence n+1 is exact, and the inclusion Ω P (log X) Ω n+1 P ( X) is a filteed quasi-isomophism. n+1 But filteed quasi-isomophisms ae quasi-isomophisms by Lemma A.4.1, so the inclusion is a quasi-isomophism, and theefoe induces isomophisms in hypecohomology by Lemma A.2.2. x k Hodge stuctues on the complement We can also conside the exact sequence degee by degee. sequence 0 Ω P n+1 Ω Pn+1(log X) Ω 1 X 0 Taking the long exact sequence in cohomology gives us an exact H q 1 (Ω p 1 X ) δ H q (Ω p P ) H q (Ω p n+1 P (log X)) Res H q (Ω p 1 n+1 X ) δ H q+1 (Ω p P ). n+1

9 THE NOETHER-LEFSCHETZ THEOREM 9 We ae pimaily inteested in the case whee p + q = n + 1, since then the middle goup is elated to H n+1 (U; C). We claim that the map Res induces an isomophism Res : H q (Ω p P (log X)) H q (Ω p 1 n+1 X ) Hn pim(x). Fist, the image of Res is the kenel of δ, which is a Hodge component of the kenel of the Gysin homomophism. We have aleady seen the kenel of the Gysin homomophism is pimitive cohomology, so it follows that the image of Res is H q (Ω p 1 X ) Hn pim(x). On the othe hand, the fist connecting map δ is the Hodge component of the Gysin homomophism, and in this case the Gysin homomophism is Poincaé dual to an isomophism by the Lefschetz hypeplane theoem, so is an isomophism. Hence the fist δ is sujective, the map H q (Ω p P ) H q (Ω p n+1 P (log X)) is the zeo map, and Res is injective. So in fact n+1 Res induces the claimed isomophism. Because the map Res : H n+1 (U) Hpim(X) n is an isomophism and Hpim(X) n = H q (Ω p 1 X ) Hn pim(x), we deduce H n+1 (U; C) = p+q=n+1 p 1 p+q=n+1 p 1 H q (Ω p P n+1 (log X)) = p+q=n+2 p,q 1 H q 1 (Ω p P n+1 (log X)). We give H n+1 (U; C) a Hodge stuctue of weight n + 2 accoding to the decomposition on the ight. This espects conjugation because the isomophism H n pim(x) = H n+1 (U) holds at the ational level, so commutes with conjugation, and hence H q 1 (Ω p P (log X)) = H p 1,q 1 n+1 pim (X) = H q 1,p 1 pim (X) = H p 1 (Ω q X ). The map Res then becomes an isomophism of Hodge stuctues of type ( 1, 1). Since the hypecohomology of the pola complex Ω P n+1 ( X) also computes the cohomology of U, we can also ty to put a Hodge stuctue on H n+1 (U) by putting a filtation on the hypecohomology H n+1 (Ω P n+1 ( X)). The pole ode filtation on Ω P n+1 ( X) induces a filtation on this hypecohomology goup. Namely, we let F p pole Hn+1 (Ω P n+1 ( X)) be the image of the map in hypecohomology induced by the inclusion of complexes F p pole Ω P n+1 ( X) Ω P n+1 ( X). Theoem 3.1. The pole ode filtation on H n+1 (U) coincides with the filtation on H n+1 (U) induced by the esidue map. Thus the esidue map induces isomophisms H 0 (Ω n+1 P ((n p + 1)X)) n+1 H 0 (Ω n+1 P ((n p)x)) + dh 0 (Ω n n+1 P ((n p)x) n+1 H p,n p (X). pim

10 10 JACK HUIZENGA Poof. We have aleady seen that the inclusion (Ω P n+1 (log X), F tiv ) (Ω P n+1 ( X), F pole ) is a filteed quasi-isomophism. Bott s vanishing theoem implies that H i (P n+1, Ω j P n+1 (kx)) = 0 fo all i, k 1, j 0, fom which it follows that the highe cohomology goups of G p F pole Ω j P n+1 ( X) all vanish. Then by Theoem A.4.2 thee is a spectal sequence 1 = H p+q (X, G p F Ω P n+1( X)) Hp+q (X, Ω P n+1( X)) = Hp+q (U; C). Since the log-pole complex is filteed quasi-isomophic to the finite pole complex, thee is an analogous isomophic spectal sequence fo log-poles. In the case p + q = n + 1, the filtation defined on H p+q (U; C) by the log-pole complex has gaded pieces given by the 1 enties. Now the enties of the E page ae subquotients of these enties, hence have no lage vecto space dimension. It follows that the enties 1 of the finite pole spectal sequence give the gaded quotients of the cohomology of H n+1 (U) with the pole filtation. This is the spectal sequence associated to the filteed complex of goups (H 0 (Ω P ( X)), F n+1 pole ), so H 0 (Ω n+1 P ((q + 1)X)) n+1 H 0 (Ω n+1 P (qx)) + dh 0 (Ω n n+1 P (qx) = Hq (Ω p P (log X)) = H p 1,n (p 1) n+1 pim (X). n+1 Then we get a Hodge stuctue of weight n + 2 by letting the (p, q)-pat of H n+1 (U) be H 0 (Ω n+1 P (qx)) n+1 H 0 (Ω n+1 P ((q 1)X)) + dh 0 (Ω n n+1 P ((q 1)X)). n+1 This definition espects conjugation since the inclusion of the log complex into the finite pole complex commutes with conjugation and identifies the Hodge components of the log complex with ou poposed Hodge components of the finite pole complex. Staightfowad computations pove the following theoem. Theoem 3.2. Any ational (n + 1)-fom ϕ on P n+1 with a pole of ode q along the hypesuface X given by F = 0 has pullback to C n+2 \ {0} given by a total degee zeo fom A F q ω n+1, whee ω n+1 is the contaction of the standad volume fom on C n+2 with the Eule vecto field E = z i z i. Thus deg A = qd n 2. The fom ϕ is in if and only if A lies in the Jacobian ideal H 0 (Ω n+1 P ((q 1)X)) + dh 0 (Ω n n+1 Pn+1((q 1)X)) j F = ( f/ z 0,..., f/ z n+1 ) C[z 0,..., z n+1 ].

11 THE NOETHER-LEFSCHETZ THEOREM 11 Letting p + q = n + 1, and putting t(p) = (n p + 1)d n 2 = deg A, we thus have an isomophism R t(p) = H p,n p pim (X), whee R = C[z 0,..., z n+1 ]/j F is the Jacobian ing. 4. Peiod maps and thei deivatives Suppose we have a family X S of compact Kähle manifolds ove a smooth complex simply connected base, with special fibe X o. The local system consisting of the pimitive cohomology H n pim(x s ) is then locally tivial; a path in the base fom o to s detemines an isomophism H n pim(x s ) = H n pim(x o ) which only depends on the homotopy class of the path elative to the endpoints. The Hodge decomposition (esp. filtation) of the fibe X s then gives a decomposition (esp. filtation) on the special fibe. The Hodge numbes of the fibes ae constant because they depend on s in an uppe-semicontinuous fashion and thei sum is constant. Now wite f p fo the dimension of F p X o. Definition 4.1. The peiod map of the family X S is the map P : S G(f 1, H n pim(x o )) G(f n, H n pim(x o )) whose pth coodinate function sends s to the isomophic image of F p H n pim(x s ) in H n pim(x o ). A fundamental esult is that the peiod map is holomophic. Anothe impotant popety is Giffiths tansvesality, which tells us that the deivative dp o of the peiod map at the base point o has a vey special fom. Recall that the tangent space to the Gassmannian G(f p, Hpim(X n o )) at the point F p Hpim(X n o ) is Hom C (F p Hpim(X n o ), Hpim(X n o )/F p Hpim(X n o ). Giffiths tansvesality tells us that in fact pth coodinate of the image of the holomophic tangent space of S at o lands in which is the same thing as Hom C (F p H n pim(x o ), F p 1 H n pim(x o )/F p H n pim(x o )), Hom C (F p H n pim(x o ), H p 1,q pim (X o)) (p 1 + q = n). Additionally, an element in the pth coodinate of the image of dp 0 esticts to the element in the (p + 1)st coodinate, so these maps all vanish on F p+1 Hpim(X n o ). Thus the image lies in Hom C (H p,q 1 pim (X o), H p 1,q pim (X o)).

12 12 JACK HUIZENGA Taken togethe, the components of the deivative of the peiod map theefoe define a map dp 0 : T o S End 1,1 H n pim(x o ). Hence the deivative of the peiod map induces an action of T o S on H n pim(x o ) by endomophisms of type ( 1, 1). Anothe way to descibe this action is in tems of the Kodaia-Spence map. This map associates to a tangent vecto v T o S a cohomology class κ(v) H 1 (X o, Θ Xo ) as follows. Shinking S if necessay, choose a local holomophic vecto field V with V o = v. Coveing the special fibe X o with small open subsets of X, we can choose holomophic lifts of V into each open set. Then on the paiwise intesections, the diffeences of the lifts poject to 0 in T o S, so the diffeences must themselves be in Θ Xo. The diffeences fom a Čech 1-cocycle, so we get a cohomology class κ(v) H1 (X o, Θ Xo ). Now we have a cup-poduct mapping H 1 (X o, Θ Xo ) H q pim (Ωp X o ) H q+1 (Θ Xo Ω p X o ), which when composed with the map induced by contaction Θ Xo Ω p X o Ω p 1 X o gives a map H 1 (X o, Θ Xo ) H q pim (Ωp X o ) H q+1 pim (Ωp 1 X o ). So v induces via the Kodaia-Spence map an endomophism of Hpim(X n o ) of type ( 1, 1). This map coesponds with the deivative of the peiod map. 5. Tautological defomations of hypesufaces In this section, we let L be the line bundle O P n+1(x o ) associated to X o, and conside the tautological family of smooth hypesufaces. The base S of this family is the locus of divisos (s) in the complete linea seies L such that the coesponding diviso X s is smooth. We ae only eally inteested in a neighbohood of the base nea the base point s o, so we do not need to woy about any monodomy issues. The tangent space to S at o is natually identified with T = H 0 (P n+1, L)/Cs 0 since S is just a subset of the pojective space PH 0 (P n+1, L). Viewing H 0 (P n+1, L) as the space of ational functions on P n+1 having at most simple poles along X o, we conside the cup poduct mapping H 0 (P n+1, L) H 0 (Ω n+1 P n+1 (kx o )) H 0 (Ω n+1 P n+1 ((k + 1)X o )). We can descibe H 0 (Ω n+1 P n+1 (kx o )) as the space of homogeneous polynomials of degee kd n 2, and if we descibe the ight hand side similaly then multiplication by

13 THE NOETHER-LEFSCHETZ THEOREM 13 the ational function G/F coesponds to multiplication by G. It theefoe maps the Jacobian ideal into itself, and we get an induced mapping H 0 (P n+1, L) H p,n p pim (X o) H p 1,n p+1 pim (X o ). The constant functions act tivially on Hva p,n p (X o ), so we aive at a multiplication map T H p,n p pim (X o) H p 1,n p+1 pim (X o ). This essentially gives us a thid desciption of the deivative of the peiod map, accoding to the following esult. Poposition 5.1. Up to some constant factos, the diagam T H p,n p pim (X o) κ 1 H 1 (X o, Θ Xo ) H p,n p pim (X o) commutes. H p 1,n p+1 pim 6. The main theoem Accoding to Poposition 5.1, the map T H p,n p pim (X o) H p 1,n p+1 pim (X o ) given by the action of the Kodaia-Spence class on pimitive cohomology is sujective wheneve the multiplication map H 0 (O P n+1(x o )) H 0 (Ω n+1 P n+1 ((n p + 1)X o )) H 0 (Ω n+1 P n+1 ((n p + 2)X o )) is sujective. We call a cohomology class α H n (X o, Q) infinitesimally fixed if it is annihilated by all the Kodaia-Spence classes. Theoem 6.1. Suppose that the multiplication maps displayed above ae sujective fo p = 1,..., n. Then infinitesimally fixed classes ae fixed. That is, infinitesimally fixed classes ae all pullbacks of cohomology classes in P n+1. Poof. We will show that an infinitesimally fixed element of H p,n p pim (X o) must be zeo. Since the action of the Kodaia-Spence class maps Hodge components to Hodge components, it follows that if an element of pimitive cohomology is infinitesimally fixed then it is zeo. We can expess an abitay cohomology class in H n (X o ) as a sum of a pimitive class and a fixed class. Since fixed classes ae infinitesimally fixed, an abitay class is infinitesimally fixed if and only if its pimitive component is zeo, i.e. if and only if it is fixed.

14 14 JACK HUIZENGA So let α H p,n p pim be infinitesimally fixed, and let, be the intesection paiing on H n (X o ). By the nondegeneacy of the intesection paiing, to show α = 0 it is enough to show that α is othogonal to H n p,p (X o ). We aleady know α is othogonal to the fixed pat of this cohomology, so it suffices to show α is othogonal to H n p,p pim (X o). So let β H n p,p pim (X o). Ou assumption that the multiplication maps ae sujective implies that β can be witten as a linea combination of tems t γ, with t T and γ a pimitive class of type (n p + 1, p 1). Fo any such tem, we have α, t γ = t α, γ = 0 since α is infinitesimally fixed. Thus α, β = 0. Coollay 6.2. Suppose n is even. Fo s outside a countable union of pope subvaieties of S, we have H m,m (X s ; Q) = im(h m,m (P n+1 ; Q) H 2m (X s ; Q)), whee the map is induced by the inclusion X s P n+1. Poof. The ight hand side is clealy contained in the left hand side fo all s, so we show the othe inclusion. It suffices to show the pimitive pat of the left hand side is contained in the ight hand side. Let S be the univesal cove of S, and conside the family as being ove S. The local system fomed by the nth pimitive cohomology goups of the fibes becomes tivial, so we can identify the cohomology goups of the diffeent fibes with one anothe. Fo α H n pim(x o ), put S α = {s S : α H m,m (X s )}. The set S α is an analytic subvaiety of S. If it equals all of S, then α is constant and hence infinitesimally fixed. Then by the theoem, α is fixed as a class on any fibe, i.e. it is the estiction of a cohomology class on P n+1. Now let U S be the complement of the images of all the pope subvaieties S α S. Suppose s U, and let β H m,m pim (X s; Q). Then we conclude S β = S, so β is fixed, and comes fom P n+1. Theoem 6.3 (Noethe-Lefschetz Theoem). Assume that d 4. Outside a countable union of pope subvaieties in the paamete space of degee d sufaces in P 3, the Picad goup of the coesponding suface X is isomophic to Z, geneated by O X (1). Hence evey cuve on X is a complete intesection. Poof. Recall that Ω 3 P 3 = O P 3( 4). The multiplication map H 0 (O P n+1(x o )) H 0 (Ω n+1 P n+1 ((n p + 1)X o )) H 0 (Ω n+1 P n+1 ((n p + 2)X o ))

15 is then the multiplication map THE NOETHER-LEFSCHETZ THEOREM 15 H 0 (O(d)) H 0 (O(d(2 p + 1) 4)) H 0 (O(d(2 p + 2) 4)). We need to show this is sujective fo p = 1, 2. In case p = 1 we have the map and p = 2 gives the map H 0 (O(d)) H 0 (O(2d 4)) H 0 (O(3d 4)), H 0 (O(d)) H 0 (O(d 4)) H 0 (O(2d 4)). So long as d 4, so that all thee spaces ae the spaces of homogeneous foms of some nonnegative degee on P 3, the maps ae obviously sujective. Thus fom the coollay we conclude that thee is a union of pope subvaieties of the paamete space of degee d sufaces in P 3 outside of which all sufaces X satisfy H 1,1 (X; Q) = im(h 1,1 (P 3 ; Q) H 2 (X; Q)). The claim is that the Picad goup of any such suface is geneated by O X (1). Conside the exponential sequence 0 Z O X O X 0. The long exact sequence in cohomology gives us 0 H 1 (O X) H 2 (X; Z) H 2 (O X ) since H 1 (O X ) is a Hodge component of H 1 (X), which is zeo by the Lefschetz hypeplane theoem. So Pic X = H 1 (O X ) is natually a subspace of H2 (X, Z). This map is compatible with the desciption of line bundles as divisos, in the sense that a diviso maps to the Poincaé dual of its fundamental homology class. Hence the image of Pic X in H 2 (X; Z) is exactly the goup of algebaic cycles. The map H 2 (X; Z) H 2 (O X ) = H 0,2 (X) is the pojection, so its kenel is the integal classes of type (2, 0) + (1, 1). All integal classes of pue type ae of type (1, 1) since they ae eal, so in fact the kenel consists of integal classes of type (1, 1). Thus an integal class is of type (1, 1) if and only if it is algebaic. In othe wods, Pic X = H 2 (X; Z) H 1,1 (X). But then evey element of Pic X is a pullback fom P 3, so Pic X = Z O X (1).

16 16 JACK HUIZENGA Appendix A. Spectal sequences A.1. Fundamental Notions. A spectal sequence consists of a sequence (E, d ), = 0, 1,... of gaded goups E = p,q Z Ep,q with homomophisms d : E p+,q +1 such that (1) d d = 0; and (2) +1 = ke d p,q / im d p,q+ 1. Obseve that d inceases the total degee n = p + q by one, so that fo fixed we have a complex (E n, d ), and E +1 computes the cohomology of this complex. A spectal sequence aises when one has a filteed complex (K, d) of goups, say with a deceasing filtation F. Setting one defines the tems Z p,q = {a F p K p+q : da F p+ K p+q+1 }, by = The diffeential d then induces d. Z p,q dz p +1,q Z p+1,q 1 1 Poposition A.1.1. (E p,q, d ) is well defined, and is in fact a spectal sequence. Poof. Fo well-definedness, we must fist show that dz p +1,q Z p+1,q 1 1 is contained in Z p,q. Fist let a Z p+1,q 1 1. Then by definition a F p+1 K p+q, and da F p+ K p+q+1. Since the filtation is deceasing, also a F p K p+q. This implies a Z p,q Z p,q. If instead a Z p +1,q+ 2 1., and Z p+1,q 1 1, then da F p K p+q, and since d 2 a = 0 F p+ K p+q+1 we see that da Z p,q. Thus dz p +1,q+ 2 1 Z p,q, and the quotient E p,q makes sense. Also we must check that thee is actually an induced map d. We stat with the diffeential d : K p+q K p+q+1. Restict it to the subgoup Z p,q of K p+q. The image of this estiction is by definition contained in F p+ K p+q+1, and is annihilated by d. Thus the image of the estiction is contained in Z p+,q +1. Now we can compose with the quotient map Z p+,q +1 E p+,q +1. We claim that the composite Z p,q E p+,q +1 factos though. Eveything in dz p +1,q+ 2 Z p+1,q 1 1 maps unde d to dz p+1,q 1 1 So we get a map d : E p,q E p+,q +1. Since d is induced fom d, we get that d 2 = 0 fo fee. 1 is killed by d since d 2 = 0. And = dz (p+) +1,(q +1)+ 2 1, which is 0 in E p+,q +1.

17 THE NOETHER-LEFSCHETZ THEOREM 17 Finally we must check that +1 = ke d p,q / im d p,q+ 1. To do this, we will explicitly identify the kenel and image in question. We fist claim that ke d p,q = Z p,q +1 + Z p+1,q 1 1 dz p +1,q Z p+1,q 1 1 Fist of all, this quotient makes sense. Indeed, dz p +1,q+ 2 1 Z+1, p,q fo if a is in Z p +1,q+ 2 1, then da is in F p K p+q and d 2 a = 0. Next we claim that the quotient is contained in the kenel. To see this it suffices to check d p,q (Z+1) p,q = 0, o dz p,q +1 dz p+1,q Z p++1,q 1. So let a Z+1. p,q Then a F p K p+q, and da F p++1 K p+q+1. Since d 2 a = 0, we see da Z p++1,q 1. So indeed, the quotient above is contained in the kenel. Fo the othe inclusion, assume that a ke d p,q, whee a Z p,q. This means that da dz p+1,q Z p++1,q 1 ; wite da = db + c especting this decomposition. Wite a = b + e fo some e, so that de = c. Then b Z p+1,q 1 1. Since a F p K p+q and b F p+1 K p+q, we have e F p K p+q, and de Z p++1,q 1, so de F p++1 K p+q+1. Thus e Z+1, p,q which shows that the kenel is contained in quotient. We claim that the image of d p,q+ 1 can be identified with = dzp,q+ 1 dz p +1,q+ 2 1 im d p,q+ 1 The quotient makes sense since Z p,q+ 1 the estiction of d on Z p,q Z p+1,q Z p+1,q 1 1 Z p +1,q But d p,q+ 1 is by definition, composed with a quotient map and factoed though a subgoup of the kenel, so its image is the extension of dz p,q+ 1 what we claimed. Theefoe ke d p,q im d p,q+ 1 = Z p,q +1 + Z p+1,q 1 1 dz p,q+ 1 We then have only to show that Z p,q +1 (dz p,q+ 1 + Z p+1,q 1 1 =. Z p,q +1 Z p,q +1 (dz p,q+ 1 + Z p+1,q 1 1 ) = dz p,q+ 1 + Z p+1,q 1. in + Z p+1,q 1 1 )., which is But Z p,q +1 contains dz p,q+ 1, and Z p,q +1 Z p+1,q 1 1 = Z p+1,q 1 ; a lies in the left hand side if and only if a is in F p+1 K p+q and da is in F p++1 K p+q+1, which ae exactly the conditions to be in Z p+1,q 1. This completes the poof. Successive quotients of the filtation F fequently occu in this setting, so we intoduce the pth gaded pat G p F (Kn ) = F p K n /F p+1 K n.

18 18 JACK HUIZENGA The fist tem in the spectal sequence is 0 = G p F (Kp+q ). Fo, by definition, Z p,q 1 = F p K p+q, so that in paticula Z p+1,q 1 1 = F p+1 K p+q, Z p+2,q 3 1 = F p+2 K p+q 1, and dz p+2,q 3 1 F p+2 K p+q since the diffeential peseves the filtation. Then 0 = Z p,q 0 /Z p+1,q 1 1 = F p K p+q /F p+1 K p+q = G p F (Kp+q ). And fo fixed p, the diffeential gives us maps G p F (Kn ) G p F (Kn+1 ). Thus we conclude that 1 = H p+q (G p F (K )). If d = 0 fo k we say that the spectal sequence degeneates at E k = E. Lemma A.1.2. If on each K n the filtation F has finite length, fo fixed (p, q), the goups E p,q emain the same fom a cetain index on. Poof. Recall that = Z p,q dz p +1,q Z p+1,q 1 1 It is enough to see that the goups Z p,q, Z p +1,q+ 2 1, and Z p+1,q 1 1 stabilize fo lage enough. But this is clea since these goups all conside filtations on eithe K p+q o K p+q 1, so that thee ae only two filtations in question. In paticula, note that if we assume the filtation stats with F 0 K n = K n and ends with F K n = 0 fo some, then fo lage enough we have = ke d F p K p+q dz 0,p+q 1 p + ke d F p+1 Z p+q. The following calculation then identifies the limit. If i : F p (K ) K is the inclusion, we define a filtation F on H n (K ) by F p H n (K ) = i H n (F p (K )) Poposition A.1.3. Fo a filtation F on K with finite length on each K n, we have = G p F H p+q (K ). Poof. The cohomology of a cochain complex is just gotten by taking kenels mod images. Maps of cochain complexes induce maps on cohomology in a covaiant fashion since the.

19 THE NOETHER-LEFSCHETZ THEOREM 19 map takes coboundaies to coboundaies. Now we can conside p and q as being fixed. Then G p F H p+q (K ) = F p H p+q (K )/F p+1 H p+q (K ) = i H p+q (F p (K ))/i H p+q (F p+1 (K )) The map i : H p+q (F p (K )) H p+q (K ) can instead be egaded as a map ke d p+q F p K p+q H p+q (K ). Then We have and similaly Theefoe which equals G p F H p+q (K ) = i (ke d p+q F p K p+q )/i (ke d p+q F p+1 K p+q ). i (ke d p+q F p K p+q ) = ke dp+q F p K p+q + im d p+q 1 im d p+q 1, i (ke d p+q F p+1 K p+q ) = ke dp+q F p+1 K p+q + im d p+q 1 im d p+q 1. G p F H p+q (K ) = ke dp+q F p K p+q + im d p+q 1 ke d p+q F p+1 K p+q + im d p+q 1, ke d p+q F p K p+q ke d p+q F p K p+q (ke d p+q F p+1 K p+q + im d p+q 1 ). Now F p K p+q F p+1 K p+q = F p+1 K p+q, and im d p+q 1 ke d p+q = im d p+q 1, so this equals ke d p+q F p K p+q ke d p+q F p+1 K p+q + im d p+q 1 F p K p+q. We claim that im d p+q 1 F p K p+q = dzp 0,p+q 1. Suppose that da is in the intesection, whee a K p+q 1. Since da F p K p+q, we get that a Zp 0,p+q 1. Convesely if a Zp 0,p+q 1, then da F p K p+q and da is in the image of d p+q 1. But then the displayed quotient equals ou ealie deived expession fo Such filtations ae called biegula filtations. One says that the spectal sequence abuts to H (K, d). This is commonly denoted. H p+q (K, d). The next esult is used to compae spectal sequences fo elated filteed complexes.

20 20 JACK HUIZENGA Lemma A.1.4. If f : K L is a filteed homomophism between complexes, thee is an induced homomophism E(f ) between the spectal sequences. If E(f ) is an isomophism fo = 0, it is an isomophism fo 0 as well. In paticula, if the filtations ae biegula, the spectal sequences abut to isomophic goups. Poof. Denote by E the spectal sequence of L, by F the filtation on l, and by Z the Z-goups associated to L. To see that we get an induced map E E, it is enough to check that f(z p,q ) Z p,q and that f(dz p +1,q Z p+1,q 1 1 ) d( Z p +1,q Z p+1,q 1 1 ). This second popety will follow fom the fist and the fact that f commutes with the diffeentials. But if a Z p,q, then a F p K p+q. So f(a) F p L p+q. Also da F p+ K p+q+1, so df(a) = f(da) F p+ L p+q+1. Theefoe f(a) Z p,q, and we get an induced map E E commuting with the diffeentials of the spectal sequence. The coespondence f E(f ) is obviously functoial. To pove the second statement, it is theefoe enough to show that if f : K K and E(f ) is the identity, then E(f +1 ) is the identity. So assume E(f ) is the identity. Fo evey p, q, the induced self-map of = Z p,q dz p +1,q Z p+1,q 1 1 is the identity. We need to pove that the induced self-map of +1 = dz p,q+ 1 Z p,q +1 + Z p+1,q 1 is the identity. So let a Z+1. p,q Since Z p,q +1 Z p,q, we know that f (a) = a. This means that f(a) a (mod dz p +1,q Z p+1,q 1 1 ), and thee ae b Z p +1,q+ 2 1 and c Z p+1,q 1 1 such that f(a) = a + db + c. Since Z p,q+ 1 Z p +1,q+ 2 1, we see that db vanishes in E+1. p,q As fo c, we have dc = d(f(a) a), fom which we deduce dc F p++1 K p+q+1 since both a and f(a) ae in Z+1. p,q So actually c Z p+1,q 1, whence f +1 (a) = a. Thus f +1 is the identity, and by induction f is the identity fo all 0. A fist quadant spectal sequence yields a paticula 5-tem long exact sequence. Poposition A.1.5. The spectal sequence fo a biegulaly filteed complex (K, d, F ) yields an exact sequence 0 E 1,0 2 H 1 (K ) E 0,1 d 2 2 E 2,0 2 H 2 (K ).

21 THE NOETHER-LEFSCHETZ THEOREM 21 Poof. Fo this to be tue we pobably have to assume that the associated spectal sequence lies in the fist quadant. Fo this to be tue, we can assume that the filtation is canonically bounded. That is, we assume that F 0 K n = K n and F n+1 K n = 0. Then if p < 0, we have p 0, so and Z p,q = {a K p+q : da F p+ K p+q+1 } = Z p+1,q 1 = 0; if instead q < 0 then Z p,q = 0 and again is canonically bounded. Since the sequence is a fist quadant sequence, E 1,0 2 = E 1,0 = g 1 F H 1 (K ) = F 1 H 1 (K )/F 2 H 1 (K ). But as the filtation is canonically bounded, F 2 K 1 = 0 and hence F 2 H 1 (K ) = 0. = 0. So assume the sequence Theefoe E 1,0 2 = F H 1 1 (K ), which is a subgoup of H 1 (K ). So we get an inclusion E 1,0 2 H 1 (K ). Clealy E 0,1 3 = E 0,1 = g 0 F H 1 (K ) = F 0 H 1 (K )/F 1 H 1 (K ) = H 1 (K )/F 1 H 1 (K ). Also E 0,1 3 = ke d 2, so clealy the image of H 1 (K ) E 0,1 2 is the kenel. We also see that the image of E 1,0 2 in H 1 (K ) is exactly the kenel of H 1 (K ) E 0,1 2. At the next stage, we have E 2,0 3 = coke d 2, and E 2,0 3 = E 2,0 = g 2 F H 2 (K ) = F 2 H 2 (K ) since F H 3 2 (K ) = 0. So we have an injection coke d 2 H 2 (K ), coinciding with the natual map E 2,0 2 H 2 (K ) factoed though the kenel, and we conclude that the whole sequence is exact. The next esult tells us that the spectal sequence of a filteed complex measues the failue of the diffeential to be compatible with the filtation. Poposition A.1.6. The spectal sequence fo a filteed complex degeneates at E 1 if and only if the deivative is stictly compatible with the filtation, i.e., F p K n im d = im d F p K n 1. Poof. Assume the deivative is stictly compatible with the filtation, and conside the quotient = Z p,q dz p +1,q Z p+1,q 1 1.

22 22 JACK HUIZENGA We see that to show d p,q = 0 amounts to showing that dz p,q dz p+1,q Z p++1,q 1. So let a Z p,q. Then da F p+ K p+q+1. So thee is some b F p+ K p+q such that da = db. If 1 we also have b F p+1 K p+q, so b Z p+1,q 1 1. Theefoe dz p,q dz p+1,q+1 1 fo 1, and the spectal sequence degeneates at E 1. Convesely assume that the sequence degeneates at E 1. Then dz p,q 1 dz p+1,q Z p+2,q 1 0 fo all 1. The inclusion im d F p K p+q 1 F p K p+q im d always holds, so let da F p K p+q be an element of the intesection. Then a K p+q 1 ; if a F p K p+q 1 then we ae done. Suppose that s is the lagest index such that a F s K p+q 1. If s < p, then a Z s,p+q s 1 1. By hypothesis we can wite da = db + c, whee b Z s+1,p+q s 2 0 and c Z s+2,p+q s 2 0. That is, b F s+1 K p+q 1 and c F s+2 K p+q. Then c = d(a b), and theefoe a b Z s,p+q s 1 2. Now using the fact that the E 2 page is degeneate, we see wite dz s,p+q s 1 2 dz s+1,p+q s Z s+3,p+q s 2 1 ; c = db + c especting this decomposition. Then witing c = d(a b b ), we get a b b Z s,p+q s 1 3. Continuing in this fashion, we ceate an element b of Z s+1,p+q s 2 0 with the popety that a b Z s,p+q s 1 fo any lage fixed numbe. Povided ou filtation is finite, this guaantees d(a b) = 0, so da = d( b). Now we can apply this pocess to b, getting elements e, f, g,... with da = d( b) = d( f) = d( g) =, each one step deepe into the filtation. The pocess stops when s = p, i.e. when we have constucted z Z p,q 1 0 satisfying da = d( z), and this demonstates the stictness of the diffeential. Recall that a double complex (living in the fist quadant) consists of a bigaded goup K, = p,q Kp,q and diffeentials d : K p,q K p+1,q and δ : K p,q K p,q+1, satisfying d 2 = δ 2 = 0, dδ = δd. The associated total complex is sk n = p+q=n Kp,q with diffeential D = d + ( 1) p δ. It has two obvious filtations, F p = p K,s F q = s q K,s, and these then induce filtations F and F on the associated total complex. associated spectal sequences ae denoted by E p,q and. The

23 THE NOETHER-LEFSCHETZ THEOREM 23 Poposition A.1.7. The E 1 and E 2 pages of this spectal sequence ae given by 1 = H q (K p,, d ) and Poof. Fo the fist equality, we have 1 = H p+q (g p F (sk ), D) 2 = H p (H q (K,, d ), d ). = H p+q ( F p sk / F p+1 sk, D) = H p+q K,s / K,s, D p +s= = H p+q (K p, p, d ) = H q (K p,, d ) p+1 +s= We can also wite H p+q (K p,, D) = H q (K p,, d ) whee D acts by d and the complex is gaded by total degee p + instead of just by. Fo the second equality, we must identify the diffeential in the E 1 page of the spectal sequence with d. By definition, 1 = Z p,q 1 DZ p,q Z p+1,q 1. 0 Now let a Z p,q 1. So a F p sk p+q, and Da F p+1 sk p+q+1. Denote by πa the pojection of a to an element of K p,q and by π a the pojection of a on F p+1 sk p+q. Then a = πa + π a, so Da = Dπa + Dπ a. But Dπa = d πa since Da F p+1 sk p+q+1. Clealy Dπ a = 0 in E p+1,q 1, since π a Z p+1,q 1 0. Theefoe Da = Dπa = d πa on Since in fact 1 is a subquotient of K p,q, this says exactly that D coincides with d unde the identification of 1 with H q (K p,, d ). Summing up what we know, the E 0 page of the spectal sequence is the bigaded goup: 0 = K p,q. The fist page E 1 computes the cohomology in the vetical diection with espect to the vetical diffeential. The hoizontal diffeentials descend to cohomology since they ae maps of chain complexes, and the second page E 2 computes the cohomology in the hoizontal diection with espect to the descended hoizontal diffeential. Recall that E p,q = G p F H p+q (sk ). That is, the, hypecohomology of K, is gaded, and the gaded pats of hypecohomology can be identified with the p, q pats of the E page. In othe wods, E n is the hypecohomology of K,, and the spectal sequence computes the hypecohomology of the double complex. 1.

24 24 JACK HUIZENGA A.2. Hypecohomology. Recall that we defined hypecohomology of the holomophic de Rham complex Ω M on a compact complex manifold M as the cohomology of the total complex associated to the Čech-de Rham complex. Suppose that we now stat out with an abitay complex (K, d) of sheaves on a topological space X, and suppose that we ae given a Leay cove U = {U i } i I of X. Recall that this means that we can compute Čech cohomology diectly fom the cove. We can now fom the double complex consisting of Čech cochains C (U, K ) with diffeential d coming fom the complex of sheaves and Čech diffeential δ. In this case the cohomology of the associated total complex by definition is the hypecohomology goup of the complex of sheaves we stated with. Without assuming the cove is Leay, we have to take a diect limit ove all efinements of the open cove U : H p (K, d) = lim U H p (sc (U, K )). To compute this hypecohomology, one uses the two spectal sequences of the double complex C (U, K ) that have E 1 -tems, and 1 = lim U H q (C (K p ), δ) = H q (X, K p ) 1 = lim U H p (C q (U, K ), d), espectively. In the fist spectal sequence the deivative comes fom d, and in the second it comes fom δ. Hence, the E 2 -tems ae given by and 2 = H p (H q (X, K ), d) 2 = H q (lim U H p (C (U, K ), d), δ). This last goup can be bette explained. Fo each q-fold intesection U i1 U iq of elements of U, thee is a facto K (U i1 U iq ) in the complex (C q (U, K ), d). In fact, the complex splits as a diect sum (C q (U, K ), d) = I =q(k (U I ), d). Cohomology of chain complexes of abelian goups commutes with diect sum, so H p (C q (U, K ), d) = I =q H p (K (U I ), d).

25 THE NOETHER-LEFSCHETZ THEOREM 25 Fo any open set U, thee is a chain complex of goups (K (U), d). Restiction maps commute with diffeentials, so induce estiction maps on cohomology. Theefoe the sheaf H p (K, d) is well-defined. What we have shown is the elation H p (C q (U, K ), d) = C q (H p (K, d)). That is, cohomology commutes with taking the Čech complex. Since cohomology commutes with diect limits, we conclude that 2 = H q (lim U C (H p (K, d)), δ) = H p (X, H p (K, d)). Fo easons that become appaent below, we intoduce the tem de Rham cohomology goups fo H p dr (X, K ) = E p,0 = H p (H 0 (X, K ), d). Any global section of K p is a 0-Čech cocycle, and thus if it is d-closed, it is a cocycle in the total complex sc (U, K ). Theefoe one gets an associated element in hypecohomology: h dr : H p dr (K ) H p (X, K ). Lemma A.2.1. If the complex of sheaves is a tivial complex with K q = 0 fo q s, then the nth hypecohomology of this complex simply coincides with H n s (X, K s ). Poof. The double complex C (U, K ) is just the Čech complex of K s with some exta zeoes. The total complex sc (U, K ) satisfies (sc (U, K )) n = C n s (U, K s ), so the hypecohomology in degee n is Čech cohomology in degee n s of K s. Poposition A.2.2. A map j : A B between complexes of sheaves induces maps between the hypecohomology goups of the complexes. If j induces an isomophism on the level of cohomology sheaves (in which case we say that j is a quasi-isomophism, it induces an isomophism in hypecohomology, because the second spectal sequences ae isomophic. Poof. Clealy j induces a map of the total complexes, so induces a map on hypecohomology. If it induces an isomophism on cohomology sheaves, then the induced map on 2 will also be an isomophism. Hence the map on E pages is also an isomophism by Lemma A.1.4, and the hypecohomology goups ae isomophic. A special case of the peceding situation aises when the complex of sheaves (K, d) we stated with in fact is an exact sequence, so that the cohomology sheaves vanish. So

26 26 JACK HUIZENGA the E 2 -tems of the second spectal sequence ae all 0 and so it abuts to 0. This then also holds fo the fist spectal sequence 1 = H q (X, K p ). Lemma A.2.3. Suppose that K is an exact complex in degees 0. Assume that H i (X, K p ) = 0 fo all p > 0 and i = 1,..., q 1. Then the deivative d q+1 induces an isomophism ke(h q (X, K 0 ) H q (X, K 1 )) H q+1 dr (K, d) = ke(d : H0 (X, K q+1 ) H 0 (X, K q+2 )) im(d : H 0 (X, K q ) H 0 (X, K q+1 )). Poof. The vanishing assumption on H i (X, K p ) = E p,i 1 implies that the diffeentials d 0,q 2,..., d 0,q q ae all zeo, so that E 0,q 2 = E 0,q 3 = = E 0,q q+1 = ke d 0,q 1 = ke(h q (X, K 0 ) H q (X, K 1 )). Now d 0,q q+1 maps E 0,q q+1 to E q+1,0 q+1. All the diffeentials d i with 2 i q that map into the (q + 1, 0) position ae zeo, and all such diffeentials going out of the (q + 1, 0) position ae zeo. So E q+1,0 2 = E q+1,0 3 = = E q+1,0 q+1 = H q+1 (H 0 (X, K ), d) = H q+1 dr (K, d). Thus the diffeential d q+1 is a mapping ke d 0,q 1 H q+1 dr (K, d). The spectal sequence has to convege in the (0, q) and (q + 1, 0) positions afte this step, since then all aows map out of the fist quadant. But we know that the hypecohomology of K vanishes, since it is an exact sequence. Theefoe the cohomology of the complex ke d 0,q H q+1 1 dr (K, d) vanishes, which says that ke d q+1 = coke d q+1 = 0. Theefoe d q+1 is an isomophism. A.3. Abstact de Rham theoems. We fist fomulate an abstact vesion of de Rham s theoem with the de Rham complex A eplaced by any complex. Theoem A.3.1 (Abstact Theoem of de Rham). Let X be a topological space, let K be a complex of sheaves on X and put F = ke{d : K 0 K 1 }. (1) If K is exact, thee is a canonical identification H p (X, F ) = H p (X, K ). (2) Suppose that H p (X, K q ) = 0 fo all q and all p > 0. Then we have an isomophism H p (X, K ) H p dr (X, K ) = H p (Γ(X, K ), d).

27 THE NOETHER-LEFSCHETZ THEOREM 27 Poof. Note that while it is assumed K is exact and gaded in nonnegative dimensions, it isn t assumed that thee is a 0 appended at the beginning. So d : K 0 K 1 does not have to be injective, and the kenel is called F. Moeove, F = H 0 (K, d), and H i (K, d) = 0. The only suviving tems in 2 theefoe have q = 0, and thee E p,0 2 = H p (X, F ). Thee is then a unique tem along each diagonal of E p,q, so in fact the hypecohomology H p (K, d) is canonically equal to H p (X, F ). Fo the second assetion, since 2 = H p (H q (X, K ), d) = 0 the fist spectal sequence degeneates at E 2. The only nonzeo enty in each diagonal is E p,0 2 = H p dr (X, K ), so the hypecohomology is H p dr (X, K ). Note that if the conditions of both pats of the theoem ae satisfied, then the theoem is the statement that the cohomology of F can be computed by taking the homology of the global sections of an acyclic esolution. Now if X = M is a diffeentiable manifold and K = A M is the de Rham complex, the last assetion is pecisely the isomophism HdR n (M) = H n (A ), which is one half of the poof of the theoem that the hypecohomology of the holomophic de Rham complex gives the usual de Rham cohomology. Anothe application is an abstact poof of the de Rham theoem. Theoem A.3.2. The de Rham isomophism H n dr (M) = H n (M; R) holds. Poof. The kenel F of d : A 0 A 1 is given by the locally constant functions on M. So we see that H n dr(m) = H n (A ) = H n (M; R). The missing elements of the poof ae the exactness of the de Rham complex, and the vanishing of H p (X, A q M ) fo all q and all p > 0. Exactness of the complex is the Poincaé lemma; that H p (X, A q M ) vanishes follows fom the existence of smooth patitions of unity. If we take the holomophic de Rham complex, this does not wok, because in geneal H p (M, Ω q M ) does not vanish fo p > 0. The holomophic Poincaé lemma still gives us exactness of the sequence of sheaves, howeve, so we can still asset H n (M, Ω M) = H n (M; C). One can also obtain diectly an identification of H n (M, Ω M ) with the de Rham goup as follows. Conside the injection Ω M A M C of the holomophic de Rham complex into the usual de Rham complex. On the level of cohomology sheaves this induces an isomophism: in H 0 we just have the sheaf of locally constant functions, and the