MAT 545: Complex Geometry Fall 2008 Notes on Lefschetz Decomposition 1 Statement Let (M, J, ω) be a Kahler manifol. Since ω is a close 2-form, it inuces a well-efine homomorphism L: H k (M) H k+2 (M), L ( [α] ) = [ω α]. Har Lefschetz Theorem: If (M, J, ω) is a compact Kahler m-manifol, the homomorphism is an isomorphism for all r 0. L r : H m r (M) H m+r (M) (1) Since ω is a (1, 1)-form, by the Hoge ecomposition theorem the above claim is equivalent to the statement that each of the homomorphisms L m (p+q) : H p,q (M) H m q,m p (M) with p+q m is an isomorphism. In the Hoge iamon, L correspons to moving up 1 step along the vertical lines; the above isomorphisms take the (p, q)-slot to its reflection about the horizontal iagonal. Thus, the Hoge iamon of a compact Kahler manifol is symmetric about the horizontal iagonal. This fact also follows from the Hoge theorem (which implies that the Hoge iamon is symmetric about the vertical iagonal) an the Koaira-Serre uality (symmetry about the center of the iamon). Furthermore, the Har Lefschetz Theorem provies a relative restriction on the numbers along each of the vertical lines in the Hoge iamon: these numbers are non-ecreasing in the bottom half an non-increasing in the top half (see (2) below). For each r =0, 1,..., m, let P H m r (M) { α H r (M): L r+1 α=0 } be the primitive cohomology of M. By the Har Lefschetz Theorem, r+1 is the smallest value of s such that the homomorphism may have a kernel; furthermore, L s : H m r (M) M m r+2s (M) H m r (M) = P H m r (M) L ( H m r 2 (M) ). 1
Thus, the Har Lefschetz Theorem implies that H k (M) = s 0 k 2s m L s( P H k 2s (M) ). (2) On the other han, (2) implies (1) immeiately. Let H H 2 (M) be the Poincare ual of [ω] an enote by M : H k (M) H l (M) H k+l 2m (M) the homology intersection prouct on M (the Poincare ual of the cup prouct). Via Poincare Duality, the Har Lefschetz Theorem is equivalent to the statement that (H M) r : H m+r (M) H m r (M), is an isomorphism for all r 0. Furthermore, η H M... H M η, }{{} r P D M ( L r (P H m r (M)) ) = P D M ( {α H m+r (M): L(α)=0} ) = { η M m r (M): H M η =0 }. Thus, the primitive k-th cohomology of M, with k m, correspons to the k-cycles that are isjoint from H; thus, it is the image of the homomorphism inuce by inclusion. H k (M V ) H k (M) The Poincare ual of the Fubini-Stuy symplectic form ω F S on P n is the hyperplane class H P n 1, since P 1 ω F S = 1 = P 1 Pn P n 1 an H 2 (P n ) is one-imensional. If M P n is a compact Kahler submanifol of imension m, P D M ( ωf S M ) = H M. Thus, the Har Lefschetz Theorem in this case is equivalent to the statement that P n r : H m+r (M) H m r (M) is an isomorphism. The primitive k-th cohomology of M, with k m, correspons to the k-cycles in M that are isjoint from H. Thus, they lie in M H P n H = C n. 2
2 Proof The Har Lefschetz Theorem is a consequence of Hoge ientities an the Hoge theorem. Let (M, J, ω) be a Kahler m-manifol an Λ: A k (M) A k 2 (M) the ajoint of the homomorphism L = ω with respect to the inner-prouct inuce by (ω, J). Denote by + : A k (M) A k (M) be the corresponing -Laplacian an let H p { α A p (M): α=0 }. Hoge Ientities: If (M, J, ω) is a Kahler m-manifol L = L, Λ = Λ, (3) LΛ ΛL = (m k) I: A k (M) A k (M). (4) Hoge Theorem: If (M, J, ω) is a compact Kahler m-manifol, the homomorphism H k H k (M), α [α], is well-efine an is an isomorphism. The secon statement is vali for any Riemannian manifol, while (4) is a point-wise statement an thus follows from a irect check for C n. The ientities (3) imply that L an Λ restrict to homomorphisms L: H k H k+2, Λ: H k H k 2. By Hoge theorem, it is sufficient to prove the analogue of the Har Lefschetz Theorem for H. From (4), we obtain the following lemma. Lemma 1 If α H k, then for all s 1 ΛL s α = L s 1( C k,s α + LΛα ) for some C k,s Z such that C k,s =0 if an only if s=m k+1. Corollary 2 If k m, α H k, an L m k+1 α=0, then (a) Λα=0; (b) L s α 0 for all s=0, 1,..., m k if α 0. 3
Proof: Both statements hol for k < 0. Suppose 0 k m an both statements are vali for all k <k. By Lemma 1, 0 = ΛL m k+1 α = L m k( C k,m k+1 α + LΛα ) = L m k+1 (Λα). Thus, Λα = 0 by the k 2, s = m k+1 case of (b). The s = 0 case of (ii) clearly hols. Suppose 1 s m k an (ii) hols for all s <s. By Lemma 1, since Λα=0 by (a) an C k,s 0. ΛL s α = L s (Λα) + C k,s L s 1 α = C k,s L s 1 α 0 if α 0, Corollary 3 For all k m, H k = ker L m k+1 H k LH k 2, H 2m k = L m k H k. Proof: (1) Suppose α H k ; then L m k+s α = 0 for some s 1. If s = 1, there is nothing to prove. Suppose s 2 an By Lemma 1, ker ( L m k+s 1 : H k H 2m k+2s 2) ker L m k+1 H k LH k 2. 0 = ΛL m k+s α = L m k+s 1( C k,m k+s α + LΛα ). Thus, C k,m k+s α + LΛα ker L m k+1 H k LH k 2 ; since C k,m k+s 0, it follows that α ker L m k+1 H k LH k 2. (2) If k = m, there is nothing to prove. Suppose k < m an the statement hols for all k > k. If α H 2m k, then L s α=0 for some s 0. If s=0, α=l m k 0. Suppose s 1 an By Lemma 1, ker ( L s 1 : H 2m k H 2m k+2s 2) L m k H k. 0 = ΛL s α = L s 1( C 2m k,s α + LΛ α ). Thus, C 2m k,s α + LΛα L m k H k ; since C 2m k,s 0, it follows that α L m k H k + LH 2m k 2. On the other han, by the k =k+2 cases of the first an secon statements, LH 2m k 2 = L m k H m k. Corollaries 2 an 3 imply the analogue of the Har Lefschetz Theorem for H, since the homomorphism in (1) is injective by the former an surjective by the latter. 4
3 Applications The Hoge theorem provies restrictions on topological an complex manifols that amit a Kahler structure. One of the stanar invariants of a topological manifol M is the k-th Betti number, h k (M) im R H k (M; R) = im C H k (M; C) = im R HeR k (M; R); the last equality hols if M amits a smooth structure. If M is a compact 2m-imensional topological manifol that amits a Kahler structure, then h 2r+1 (M) is even for all r Z an h 2r (M)>0 for all r = 0, 1,..., m. Furthermore, H 2 (M; R) contains an element α such that α m 0. If (M, J) is a compact complex manifol that amits a compatible Kahler structure, then h p,q (M) im C H p,q h p,q (M) = h q,p (M) p, q, (M) = im C H p,q (M) p, q, h k (M) = p+q=r h p,q (M) r. Furthermore, the homology class of any analytic subvariety in (M, J) is non-zero in the homology of M, as is every holomorphic p-form in H p (M). The Har Lefschetz Theorem provies aitional restrictions. If M is a compact 2m-imensional topological manifol that amits a Kahler structure, then H 2 (M; R) contains an element α such that the homomorphisms H m r (M) H m+r (M), β α r β, r 0, are isomorphisms. If (M, J) is a compact complex m-manifol that amits a compatible Kahler structure, then H 1,1 (M) H2 (M; R) contains an element α such that the homomorphisms are isomorphisms. H p,q (M) Hm q,m p (M), β α m p q β, p+q m, If M is a compact topological oriente 4k-imensional manifol, the pairing Q: H 2k (M; R) H 2k (M; R) R, α β α β, [M]. is non-egenerate by Poincare uality an is symmetric. Thus, H 2k (M; R) amits a basis with respect to which this pairing is iagonal with each of the non-zero entries equal to +1 or 1. Let λ ± (M)=λ ± (Q) enote the number of ±1 entries; this number is etermine by the bilinear form Q an thus by the topology an the orientation of M. So, is the number σ(m) = λ + (M) λ (M), which is known as the signature of M. If in aition J is a complex structure on M, Q restricts to a non-egenerate symmetric pairing Q k,k on H k,k (M) H 2k (M; R). Let λ p,p ± (M)=λ ±(Q p,p ). 5
Inex Theorem for Surfaces: If (M, J) is a compact connecte complex surface (im C M = 2) that amits a compatible Kahler structure, then λ 1,1 + (M)=1. Proof: Let ω be a symplectic form on M compatible with J an H 1,1 the corresponing space of harmonic (1, 1)-forms. By the Har Lefschetz Theorem, H 1,1 = Cω { α H 1,1 : ω α=0 } V + V 0. This ecomposition is Q-orthogonal, an Q restricte to V + H 2 (M; R) is positive-efinite. Thus, it is sufficient to show that Q(α, α) 0 α V 0 H 2 (M; R). Given p M, let (z 1, z 2 ) be holomorphic coorinates aroun p on M so that ω p = 1 2 Im( z 1 z 1 + z 2 z 2 ) p = x 1 y 1 + x 2 y 2, where z j =x j +iy j. If α H 1,1 H 2 (M; R), α p = ( A x 1 y 1 + B x 2 y 2 + C(x 1 x 2 + y 1 y 2 ) + D(x 1 y 2 y 1 x 2 ) ) p for some A, B, C, D R. If in aition ω α = 0, A= B an α p α p = 2(A 2 +C 2 +D 2 )x 1 y 1 x 2 y 2 p. Thus, α p α p is a non-positive multiple of the volume form on M for all p M an thus Q(α, α) 0 as neee. If (M, J) is a compact complex 2k-manifol that amits a compatible Kahler structure, then σ(m) = p+q 0(mo 2) h p,q (M). This is euce from the Hoge-Riemann bilinear relations (a single formula) in Griffiths&Harris, pp123-126. As explaine in the top half of p124, these relations follow from the Har Lefschetz Theorem an quite a bit of representation theory (a classical subject). There is an important typo on p123, in the line before the statement of the Hoge-Riemann bilinear relations: k = p + q shoul in fact be n k = p + q. 6