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Advances in Mathematics 180 (2003) 87–103

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Harmonic cohomology of symplectic manifolds Stefan Haller1 Institute of Mathematics, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria Received 2 January 2002; accepted 18 November 2002 Communicated by Tomasz S. Mrowka

Abstract Mathieu (Math. Helv. 70 (1995) 1) introduced a canonic ﬁltration in the de Rham cohomology of a symplectic manifold and proved, that the middle ﬁltration space is the space of harmonic cohomology classes. We give an interpretation of the other ﬁltration spaces, prove a Ku¨nneth theorem for harmonic cohomology, prove Poincare´ duality for harmonic cohomology and show how surjectivity of certain Lefschetz type mappings is related to properties of the ﬁltration. For a closed P symplectic manifold M we also introduce symplectic invariants sl ðMÞ; lX0; and show lX0 sl ðMÞ ¼ signðMÞ if M is of dimension 2n with n even. r 2003 Elsevier Science (USA). All rights reserved. MSC: Primary 53D35; Secondary 58A12; 58A14; 57R17 Keywords: Symplectic manifolds; Harmonic cohomology; Signature

1. Introduction and main results Let ðM; oÞ be a symplectic manifold of dimension 2n: It need not be compact and might have non-trivial boundary. Libermann introduced a star operator * : Onk ðMÞ-Onþk ðMÞ; satisfying * 2 ¼ 1: Setting d : Ok ðMÞ-Ok1 ðMÞ; da :¼ ð1Þkþ1 * d * a; one is led to the subspace H0k ðMÞDH k ðMÞ of harmonic cohomology classes, i.e. those having representatives a with da ¼ da ¼ 0: Throughout the paper H n ðMÞ will always denote de Rham cohomology. Brylinski [B88] E-mail address: [email protected] S. Haller is supported by the ‘Fonds zur Fo¨rderung der wissenschaftlichen Forschung’ (Austrian Science Fund), project number P14195-MAT. 1

0001-8708/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0001-8708(02)00083-X

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proved, that every de Rham cohomology class on a closed Ka¨hler manifold has a harmonic representative, and conjectured that this was true for arbitrary closed symplectic manifolds. Recall, that a symplectic manifold is said to satisfy the Hard Lefschetz Theorem if the cup product ½ok : H nk ðMÞ-H nþk ðMÞ is onto, for all kX0: Mathieu showed, see [M95], that a symplectic manifold satisﬁes the Hard Lefschetz Theorem if and only if every cohomology class has a harmonic representative, and provided counter examples to the Brylinski conjecture. Let b denote the real two-dimensional Lie algebra with base fh; eg and structure ½h; e ¼ 2e: Note, that H n ðMÞ is a b-module via e½a ¼ ½o4½a and h½a ¼ ðk nÞ½a; for ½aAH k ðMÞ: Mathieu [M95] showed that a large class of b-modules including H n ðMÞ; admit a canonic ﬁltration by b-submodules, see Proposition 2.5. Let us write H n ðMÞm for this ﬁltration. Mathieu then proves, that the space of harmonic cohomology classes H0n ðMÞ is precisely H n ðMÞ0 : Particularly the space of harmonic classes does only depend on ½oAH n ðMÞ: In Section 4, see Deﬁnition 4.3, we will introduce a ﬁltration Hmn ðMÞ of H n ðMÞ; by asking the cohomology classes to satisfy certain harmonicity conditions. It then turns out, that this ﬁltration is the same as the ﬁltration H n ðMÞm : More precisely we have Theorem 1.1. Suppose ðM; oÞ is a symplectic manifold of dimension 2n and mAZ: Then Hmn ðMÞ ¼ H n ðMÞm : Particularly the filtration Hmn ðMÞ does only depend on ½oAH n ðMÞ: If ðM 0 ; o0 Þ is another symplectic manifold of dimension 2n0 and k g : M 0 -M a smooth map, such that gn ½o ¼ ½o0 AH 2 ðM 0 Þ; then gn maps Hmn ðMÞ k 0 to Hmn0 ðM Þ: k Set H˜ km ðMÞ :¼ Hmk ðMÞ=Hm1 ðMÞ; and if the cohomology is ﬁnite dimensional, let k k b ðMÞ :¼ dim H ðMÞ; bm ðMÞ :¼ dim Hmk ðMÞ; b˜km ðMÞ :¼ dim H˜ km ðMÞ and deﬁne P k b ðMÞtk ; pM the corresponding Poincare´ polynomials pM ðtÞ :¼ m ðtÞ ¼ P P P k k M k k M M ˜ bm ðMÞt : Note, that p ¼ m p˜ m and for the harmonic bm ðMÞt ; p˜ m ðtÞ :¼ P Poincare´ polynomial we get pM ˜M 0 ¼ m: mp0 p k

Theorem 1.2. Suppose ðM; oÞ is a symplectic manifold of dimension 2n and mAZ: Then Hmk ðMÞ ¼ 0 for kX2n þ 2m þ 1 and Hmk ðMÞ ¼ H k ðMÞ for kp2m þ 1: For ðMÞ-H˜ nþmþk ðMÞ is an isomorphism. If H n ðMÞ is finite kX0; ½ok : H˜ nþmk m m 2nþ2m M dimensional we thus have p˜ M p˜ m ð1=tÞ: Moreover, setting rij :¼ m ðtÞ ¼ t rankð½oj : H i2j ðMÞ-H i ðMÞÞ we have bnþmk ðMÞ bnþmk ðMÞ ¼ rnþmþk bnþmþk ðMÞ m m k X ¼ rnþmþkþ2l rnþmþkþ2l : kþ2l1 kþ2l lX1

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The following tells how the property Hmn ðMÞ ¼ H n ðMÞ is related to the surjectivity of certain Lefschetz-type mappings. For m ¼ 0 this is Mathieu’s result, that a symplectic manifold satisﬁes the Hard Lefschetz Theorem if and only if every de Rham cohomology class has a harmonic representative, cf. [M95]. Theorem 1.3. Let ðM; oÞ be a symplectic manifold of dimension 2n; mAZ and kX0: Then the following are equivalent: (1) For all lX0 we have H nþmþkþ2l ðMÞ ¼ Hmnþmþkþ2l ðMÞ: (2) For all lX0 we have H nþmþkþ2l ðMÞ ¼ Hmnþmþkþ2l ðMÞ and moreover H nþmk2lþ2 ðMÞ ¼ Hmnþmk2lþ2 ðMÞ: (3) For all lX0 the map ½okþ2l : H nþmk2l ðMÞ-H nþmþkþ2l ðMÞ is onto. Particularly, Hmn ðMÞ ¼ H n ðMÞ iff ½ol : H nþml ðMÞ-H nþmþl ðMÞ is onto for all lX0: Theorem 1.4. Suppose M is a closed symplectic manifold of dimension 2n and m; kAZ: Then the well-defined bilinear pairing Z ˜ nþk ðMÞ-R; ð½a; ½bÞ :¼ ðMÞ# H a4b ð; Þ : H˜ nk m m M

2n M ˜ m ð1=tÞ: If n is even we moreover have is non-degenerate, that is p˜ M m ðtÞ ¼ t p

signðMÞ ¼ signðð; Þ : H˜ n0 ðMÞ#H˜ n0 ðMÞ-RÞ; where sign denotes the signature. Example 1.5. Suppose M is a closed symplectic manifold, which satisﬁes the Hard Lefschetz Theorem. By Theorem 1.3 Hmn ðMÞ ¼ H n ðMÞ; for all mX0: Theorem 1.4 now implies Hmn ðMÞ ¼ 0; for all mo0; that is a cohomology class which has a coexact representative vanishes. Remark 1.6. In Remark 4.4 we will use the pairing H˜ n0 ðMÞ#H˜ n0 ðMÞ-R to deﬁne symplectic invariants P sl ðMÞ; lX0; for any closed symplectic manifold M: If n is even we will see, that lX0 sl ðMÞ ¼ signðMÞ: Recall the well-known fact, that on a closed symplectic manifold, which satisﬁes the Hard Lefschetz Theorem, all odd degree Betti numbers are even. The following can be thought of generalization to arbitrary closed symplectic manifolds. Theorem 1.7. Suppose M is a closed symplectic manifold of dimension 2n: Then the well-defined bilinear pairing Z Z on 0; T : H˜ k0 ðMÞ#H˜ k0 ðMÞ-R; 0½a; ½bT :¼ a4 * b ¼ /a; bS n! M M

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is non-degenerate. It is symmetric for k even and skew-symmetric for k odd. Particularly b˜k0 ðMÞ is even for odd k: The paper is organized as follows. In Sections 2 and 3 we present the algebraic preparations. In Section 4 we discuss the application to the space of forms on a symplectic manifold and prove the theorems stated above. We also present a Ku¨nneth theorem, see Proposition 4.5. Section 5 then is dedicated to a more involved example, where we compute the harmonic Betti numbers of CPn blown up along a closed codimension 2k symplectic submanifold, see Theorem 5.2.

2. Canonic ﬁltration on b-modules Let g :¼ slð2; RÞ with base fe; f ; hg and relations ½h; e ¼ 2e; ½h; f ¼ 2f ; ½e; f ¼ h: Let h denote the subalgebra spanned by h; and b the subalgebra spanned by fe; hg: Let Vh denote the category of h-modules V ; which admit a decomposition V ¼ "kAZ V k into eigenspaces of h; V k being the eigenspace to the weight k; and only ﬁnitely many V k non-trivial. Moreover, let Vb resp. Vg denote the category of b resp. g-modules for which the underlying h-module is in Vh : Then e : V k -V kþ2 and f : V k -V k2 :2 If we talk about an h; b or g-module we will always mean modules in Vh ; Vb ; Vg ; respectively, and similar for submodules. So submodules are graded submodules, with respect to the decomposition into eigenspaces of h: The next lemma can be found in [Y96]. Lemma 2.1. Suppose V AVg ; kX0; lAZ and vAV l : (1) (2) (3) (4)

The g-module generated by v is finite dimensional. The mappings ek : V k -V k and f k : V k -V k are isomorphisms. If l þ kX0 then ek v ¼ 0 if and only if f lþk v ¼ 0: Set PV :¼ fvAV : fv ¼ 0g: Then V ¼ "kX0 ek PV :

Proof. Consider the subspace W DV ; spanned by fej f i v: i; jX0g: Then W is ﬁnite dimensional since V AVg and it is a g-submodule for we have ½ej ; f ¼ jhej1 þ jð1 jÞej1 : This proves (1). The remaining assertions follow from ﬁnite-dimensional representation theory and (1). & For kAZ we let Rk AVb denote the one-dimensional b-module with base fzg and action hz :¼ kz; ez :¼ 0: 2

If V is a g-module and one just assumes, that it admits a decomposition into ﬁnitely many eigenspaces of h; then it follows that all weights are integers, cf. [Y96] or Lemma 2.1(1).

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Lemma 2.2. Suppose V ; W AVg and k; lAZ: Then: (1) If j : V #Rl -W #Rk is a non-zero b-module homomorphism then lXk: (2) If j : V -W is a b-module homomorphism then j is a g-module homomorphism. Proof. We start proving (1). Tensoring with Rl we may assume l ¼ 0: Choose vAPV j ; such that jðvÞa0; cf. Lemma 2.1(4), and choose wAW jk with jðvÞ ¼ w#z: From Lemma 2.1(3) we get ejþ1 v ¼ 0; hence 0 ¼ jðejþ1 vÞ ¼ ejþ1 jðvÞ ¼ ðejþ1 wÞ#z; and so ejþ1 w ¼ 0: If k þ jo0 then ko0 and we are done. If k þ jX0 then by Lemma 2.1(2) ekþj oa0; but since ejþ1 o ¼ 0 we must have j þ 14k þ j; i.e. kp0: To see (2) note ﬁrst, that if jð fvÞ ¼ f jðvÞ for some v then the same relation holds for v replaced by ev: Indeed jð fevÞ ¼ jðefv hvÞ ¼ ejð fvÞ hjðvÞ ¼ ef jðvÞ hjðvÞ ¼ fejðvÞ ¼ f jðevÞ: In view of Lemma 2.1(4) it remains to show jðPV ÞDPW : To see this let vAPV k : Lemma 2.1(3) then gives ekþ1 v ¼ 0; so ekþ1 jðvÞ ¼ jðekþ1 vÞ ¼ 0; hence f jðvÞ ¼ 0; again by Lemma 2.1(3). & For V AVb we write V AVg if the b-module structure extends to a g-module structure. The latter then is unique by Lemma 2.2(2). p

Lemma 2.3. Suppose U; V AVb ; W AVg ; kAZ; U-V - W #Rk a short exact sequence of b-modules and suppose ej : U kj -U kþj is onto, for all jX0: Then the sequence splits, i.e. V CU"ðW #Rk Þ: Proof. Tensoring with Rk we may assume k ¼ 0: For wAPW j we ﬁnd vAV j with pðvÞ ¼ u: From Lemma 2.1(3) we get ejþ1 vAU and so we ﬁnd uAU j2 with ejþ2 u ¼ ejþ1 v: Therefore w/v eu extends uniquely to a b-module homomorphism /wSb -V ; where /wSb denotes the b-submodule generated by w: Applying this construction to a base of PW the lemma now follows from Lemma 2.1(4). & Lemma 2.4. Suppose 0aV AVb ; let n resp. nþ denote the smallest resp. largest integer, such that V n a0 resp. V nþ a0 and let LðV Þ denote the set of integers l for which there exists a non-zero submodule W DV with W #Rl AVg : Finally set lðV Þ :¼ min LðV Þ; cf. (1). Then: (1) LðV Þa|; LðV ÞDfn ; y; nþ g: (2) There is a maximal submodule VlðV Þ DV with VlðV Þ #RlðV Þ AVg : (3) If V =VlðV Þ a0 then lðV =VlðV Þ Þ4lðV Þ: Proof. Certainly V nþ #Rnþ AVg ; thus nþ ALðV Þ and LðV Þa|: If 0aW DV and W #Rl AVg then W #Rl has a weight p0 and one X0; so W has a weight pl and one Xl and hence n plpnþ : This proves (1). Part (2) follows from Zorn’s lemma and Lemma 2.2(2). Set l :¼ lðV Þ and p : V -V =Vl : To see (3) we ﬁrst rule out l ¼ lðV =Vl Þ: If this were the case we found W DV =Vl ; W #Rl AVg : By Lemmas 2.3 and 2.1(2) the short

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exact sequence Vl -p1 ðW Þ-W splitted, which would contradict the maximality of Vl : Remains to rule out l4lðV =Vl Þ: If this were the case we found vAV lk ; veVl ; ek vAVl : So there existed v*AVllk with ek v* ¼ ek v; i.e. ek ðv v*Þ ¼ 0; a contradiction to l ¼ lðV Þ ¼ min LðV Þ: So we must have lðV =VlðV Þ Þ4lðV Þ: & The next proposition establishes Mathieu’s ﬁltration and some properties we’ll need later, cf. [M95]. Proposition 2.5. Let V AVb : Then: (1) There exists a unique filtration by b-submodules ?DVm1 DVm D? of V ; such that Vm ¼ 0 for m sufficiently small, Vm ¼ V for m sufficiently large and such that ðVm =Vm1 Þ#Rm AVg ; for all mAZ: (2) If W AVb is another b-module with corresponding filtration Wm and j : V -W a b-module homomorphism, then jðVm ÞDWm : (3) V C"mAZ Vm =Vm1 and Vm ¼ "mpm Vm˜ =Vm1 : ˜ ˜ Proof. The existence of a ﬁltration as in (1) follows from Lemma 2.4. Suppose W AVb ; Wm a ﬁltration as in (1) and j : V -W a b-module homomorphism. We are going to show jðVm ÞDWm by induction on m: For m sufﬁciently small this is trivial. Assume inductively jðVm1 ÞDWm1 and let j be the smallest integer, such that jðVm ÞDWj : We have to show jpm: Assume conversely j4m: Then j : Vm =Vm1 -Wj =Wj1 is a well-deﬁned non-zero b-module homomorphism. From Lemma 2.2(1) we get mXj; a contradiction to j4m; hence jðVm ÞDWm : Particularly a ﬁltration as in (1) is unique. To see (3) it sufﬁces to show, that the short exact sequence Vm1 -Vm -Vm =Vm1 splits, but this follows from Lemma 2.3 and Proposition 2.6(2). & For V AVb we set V˜ m :¼ Vm =Vm1 AVb : Proposition 2.6. Let V AVb ; mAZ and kX0: Then: (1) (2) (3) (4)

˜ mþk is an isomorphism. ek : V˜ mk m -Vm k mk e : Vm -Vmmþk is onto. ekþ1 : V mk =Vmmk -V mþkþ2 =Vmmþkþ2 is injective. If V is finite dimensional and rij :¼ rankðej : V i2j -V i Þ; then dim V mk dim Vmmk ¼ rmþk dim Vmmþk k X mþkþ2l ¼ rmþkþ2l : kþ2l1 rkþ2l lX1

Proof. Statement (1) follows from Lemma 2.1(2). We prove (2) by induction on m: For m sufﬁciently small Vm ¼ 0 and the statement is trivial. Inductively, we assume

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mþk mþk mk2 mk ekþ1 : Vm1 -Vm1 and hence ek : Vm1 -Vm1 is surjective. From (1) we see, mþk k mk mk mþk that e : Vm =Vm1 -Vm =Vm1 is onto. Thus ek : Vmmk -Vmmþk has to be surjective, too. We prove (3) by induction on m: For m sufﬁciently large Vm ¼ V and the mþkþ4 mk -V mþkþ4 =Vmþ1 and statement is trivial. Inductively, assume ekþ2 : V mk =Vmþ1 mþkþ2 kþ1 mk mk mþkþ2 kþ1 hence e :V =Vmþ1 -V =Vmþ1 is injective. From (1) we see, that e : mþkþ2 mk mk mþkþ2 kþ1 mk mk mþkþ2 Vmþ1 =Vm -Vmþ1 =Vm is injective. Thus e :V =Vm -V = Vmmþkþ2 is injective as well. Next we show (4). From (3) we get kerðek : Vmmk -Vmmþk Þ ¼ kerðek : V mk -V mþk Þ; and thus from (2)

dim Vmmk ¼ dim Vmmþk þ dim V mk rmþk ; k

ð2:1Þ

the ﬁrst equality. From (2) we get imgðekþ1 : V mk -V mþkþ2 Þ+Vmmþkþ2 and thus from (3) dim V mk dim Vmmk ¼ rmþkþ2 dim Vmmþkþ2 : kþ1

ð2:2Þ

Adding Eqs. (2.1) and (2.2), the ﬁrst for k þ 2; we obtain dim V mk dim Vmmk ¼ ðdim V mk2 dim Vmmk2 Þ þ ðrmþkþ2 rmþkþ2 Þ kþ1 kþ2 and thus dim V mk dim Vmmk ¼

P

lX1

mþkþ2l rmþkþ2l : kþ2l1 rkþ2l

&

Proposition 2.7. Suppose V AVb ; mAZ and kX0: Then the following are equivalent: (1) For all lX0 we have V mþkþ2l ¼ Vmmþkþ2l : (2) For all lX0 we have V mþkþ2l ¼ Vmmþkþ2l and V mk2lþ2 ¼ Vmmk2lþ2 : (3) For all lX0 the map ekþ2l : V mk2l -V mþkþ2l is onto. Particularly Vm ¼ V iff el : V ml -V mþl is onto for all lX0: Proof. Implication ð2Þ ) ð1Þ is trivial. Implication ð1Þ ) ð2Þ follows from Proposition 2.6(3). Implication ð1Þ ) ð3Þ is a consequence of Proposition 2.6(2). To see ð3Þ ) ð1Þ note ﬁrst, that part (2) and (3) of Proposition 2.6 give, for jX0; V mþjþ2 ¼ Vmmþjþ2

)

ej V mj ¼ Vmmþj :

Using that one proves ð3Þ ) ð1Þ by induction on l: For l sufﬁciently large the statement is trivial, since V mþkþ2l ¼ 0: For the last assertion apply the previous for k ¼ 0 and k ¼ 1: &

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One easily veriﬁes the following Lemma 2.8. Suppose V ; W AVb : Then: (1) ðV n Þ ¼ faAV n : aj fn ˜ Vm1 ¼ 0g and ðV Þm CVm #R2m : m g Þ CV˜ m "W (2) ðV "W Þ ¼ Vm "Wm and ðV"W ˜ m: m m P (3) ðV #W Þm ¼ m1 þm2 ¼m Vm1 #Wm2 and moreover g Þ C"m þm ¼m V˜ m1 #W ˜ m2 : ðV#W m

1

2

3. Harmonic cohomology classes of a-modules Let a :¼ g R2 denote the semi-direct product, where g acts on the Abelian Lie algebra R2 in the standard way. Choose a base fd; dg of R2 ; such that hd ¼ d; ed ¼ 0; fd ¼ d; hd ¼ d; ed ¼ d and f d ¼ 0: We consider a as a Z-graded Lie algebra a ¼ a2 "a1 "a0 "a1 "a2 ; with base ff ; d; h; d; eg: Then the graded commutators are ½h; e ¼ 2e; ½h; f ¼ 2f ; ½e; f ¼ h; ½h; d ¼ d; ½h; d ¼ d; ½e; d ¼ 0; ½ f ; d ¼ 2 2 d; ½e; d ¼ d; ½ f ; d ¼ 0; ½d; d ¼ 2d ¼ 0; ½d; d ¼ 2d ¼ 0 and ½d; d ¼ dd þ dd ¼ 0: Let Va denote the category of a-modules, such that the underlying h-module is in Vh : Then f ; d; h; d; e act by endomorphism of degree 2; 1; 0; 1; 2; respectively, on the graded vector space V ¼ "kAZ V k : For V AVa we deﬁne HðV Þ :¼ ðker dÞ=ðimg dÞ; it inherits a b-module structure from V : Deﬁnition 3.1. Suppose V AVa and kAZ: We let Z0k denote the space of vAV k for k DV k denote the space of vAV k ; such that which dv ¼ dv ¼ 0: For m40 we let Zm dv ¼ 0 and such that there exist vj AV k2j ; 1pjpm; satisfying dv ¼ dv1 ; dvj ¼ k denote the space of vAV k ; such dvjþ1 ; 1pjpm 1 and dvm ¼ 0: For mo0 we let Zm kþ2j1 ; 1pjp m; satisfying v ¼ dv1 ; that dv ¼ 0 and such that there exist vj AV k k dvj ¼ dvjþ1 ; 1pjp m 1: Finally set Hmk ðV Þ :¼ Zm =ððimg dÞ-Zm ÞDH k ðV Þ: Obviously Zm DZmþ1 and thus ?DHm ðV ÞDHmþ1 ðV ÞD? is a ﬁltration of the vector space HðV Þ: Note, that H0 ðV Þ is the space of harmonic cohomology classes, i.e. those having harmonic representatives. Let n be the largest integer, such that V n a0; equivalently the smallest integer, such that V n a0; cf. Lemma 2.1(2). k Lemma 3.2. Suppose mX0; vAV k ; dv ¼ 0; f mþ1 v ¼ 0: Then vAZm : Particularly k k k k Zm ¼ ðker dÞ-V and hence Hm ðV Þ ¼ H ðV Þ for kp n þ 2m þ 1:

Proof. Inductively one checks ½ f k ; d ¼ kdf k1 ¼ kf k1 d:

ð3:3Þ

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Now set vj :¼ ð1Þj j!1 f j v; 1pjpm þ 1: Using (3.3) one checks dv ¼ dv1 ; dvj ¼ dvjþ1 for 1pjpm and hence dvm ¼ dvmþ1 ¼ 0 since vmþ1 ¼ 0: & k Lemma 3.3. Suppose mo0; vAZm and vj ; 1pjp m; as in Definition 3.1. k Dðimg dÞ-V k and hence If in addition dvm ¼ 0 then v is exact. Particularly Zm Hmk ðV Þ ¼ 0 for kXn þ 2m þ 1: 1 f j1 dvj1 for 2pjp m Proof. Indeed from (3.3) we get dðj!1 f j vj Þ ¼ j!1 f j dvj ðj1Þ! and thus

d

1 1 f m vm fv1 þ f 2 v2 þ ? þ 2 ðmÞ!

¼ dfv1 fdv1 ¼ dv1 ¼ v:

&

k kþ2 Lemma 3.4. Suppose mX0 and vAZm : Then evAZm :

Proof. For m ¼ 0 this is obvious, so suppose m40: Choose vj ; 1pjpm; as in Deﬁnition 3.1, set v01 ¼ ev1 þ mv; v0j :¼ evj þ ðm j þ 1Þvj1 ; 2pjpm; and check dðevÞ ¼ 0; dðevÞ ¼ dv01 ; dv0j ¼ dv0jþ1 ; 1pjpm 1; dv0m ¼ 0: & k Lemma 3.5. Suppose mo0; vAZm and vj ; 1pjp m; as in Definition 3.1. Then kþ2 ev þ mdv1 AZm :

Proof. Set v0j :¼ evj þ ðm þ jÞvjþ1 ; 1pjp m 1; and v0m :¼ evm : Then check dv01 ¼ ev þ mdv1 and dv0j ¼ dv0jþ1 ; 1pjp m 1: & Lemmas 3.4 and 3.5 show, that Hm ðV Þ is a b-submodule of HðV Þ: Deﬁne H˜ m ðV Þ :¼ Hm ðV Þ=Hm1 ðV Þ; and let Em : H˜ km ðV Þ-H˜ kþ2 m ðV Þ denote the mapping induced k k ˜ ˜ by e: Moreover let Am : Hm ðV Þ-Hm ðV Þ be the mapping given by multiplication by k m: We are going to deﬁne Fm : H˜ km ðV Þ-H˜ k2 m ðV Þ; such that the operators Em ; Fm ; and Am make H˜ m ðV Þ into a g-module. We need another two easy lemmas for that. k Lemma 3.6. Suppose m40; vAZm and vj ; 1pjpm; as in Definition 3.1. Then k2 : fv þ v1 AZm

Proof. Set v0j :¼ fvj þ ðj þ 1Þvjþ1 ; 1pjpm 1; and v0m :¼ fvm : Then check dð fv þ v1 Þ ¼ 0; dð fv þ v1 Þ ¼ dv01 ; dv0j ¼ dv0jþ1 ; 1pjpm 1 and dv0m ¼ 0: & k k2 Lemma 3.7. Suppose mp0 and vAZm : Then fvAZm :

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Proof. For m ¼ 0 this is obvious, so suppose mo0: Choose vj ; 1pjp m; as in Deﬁnition 3.1, set v01 :¼ fv1 ; v0j :¼ fvj ðj 1Þvj1 ; 2pjp m; and check dfv ¼ 0; dv01 ¼ fv as well as dv0j ¼ dv0jþ1 ; 1pjp m 1: & It follows easily from Lemmas 3.6 and 3.7, that we have well-deﬁned operators ( Fm ½v :¼ ½ fv þ v1 ; m40; k k2 Fm : H˜ m ðV Þ-H˜ m ðV Þ; Fm ½v :¼ ½ fv; mp0; k where v is a representative in Zm :

Lemma 3.8. For mAZ the operators Em ; Fm ; and Am make H˜ m ðV Þ into a g-module, that is ½Am ; Em ¼ 2Em ; ½Am ; Fm ¼ 2Fm ; ½Em ; Fm ¼ Am : Proof. The ﬁrst two commutator relations are obvious. We check the third one for k m40 ﬁrst. Suppose vAZm and choose vj ; 1pjpm; as in Deﬁnition 3.1. Then ½Em ; Fm ð½vÞ ¼ Em Fm ð½vÞ Fm Em ð½vÞ ¼ Em ð½ fv þ v1 Þ Fm ð½evÞ ¼ ½efv þ ev1 ½ fev þ ev1 þ mv ¼ ½kv mv ¼ Am ð½vÞ: Now suppose mo0 and choose vj ; 1pjp m; as in Deﬁnition 3.1. Then ½Em ; Fm ð½vÞ ¼ Em Fm ð½vÞ Fm Em ð½vÞ ¼ Em ð½ fvÞ Fm ð½ev þ mdv1 Þ ¼ ½efv þ mdfv1 ½ fev þ mfdv1 ¼ ½ðef feÞv mð fd df Þv1 ¼ ½kv mdv1 ¼ ½kv mv ¼ Am ð½vÞ: For m ¼ 0 the relation ½Em ; Fm ¼ Am is proved similarly.

&

Proposition 3.9. For V AVa and mAZ we have Hm ðV Þ ¼ HðV Þm : Proof. Hm ðV Þ is a ﬁltration of HðV Þ by b-submodules. For sufﬁciently small m we have Hm ðV Þ ¼ 0 by Lemma 3.3, for m sufﬁciently large we have Hm ðV Þ ¼ HðV Þ by Lemma 3.2, and from Lemma 3.8 we get ðHm ðV Þ=Hm1 ðV ÞÞ#Rm AVg : So

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the proposition follows Proposition 2.5(1). &

97

from the uniqueness of such a ﬁltration, see

4. Application to symplectic manifolds Suppose ðM; oÞ is a symplectic manifold of dimension 2n: Let x1 :¼ w : XðMÞ-O1 ðMÞ the usual isomorphism given by wX :¼ iX o; and extend them to graded algebra isomorphism x1 ¼ w : Xn ðMÞ-On ðMÞ; i.e. xða4bÞ ¼ ðxaÞ4ðxbÞ: Here Xn ðMÞ denotes the graded algebra of skew-symmetric multi-vector ﬁelds. Next extend the contraction of vector ﬁelds with forms to i : Xl ðMÞ#Okþl ðMÞ-Ok ðMÞ; n such that iX 4Y a ¼ iY iX a: The symplectic star operator is deﬁned by * a :¼ ixa on! ; it was ﬁrst considered by Libermann in her thesis. It satisﬁes * 2 ¼ 1: For example one n on1 for a 1-form a: If one extends the bilinear form o to has * 1 ¼ on! and * a ¼ a4ðn1Þ! Lk T n M one ends up with a non-degenerate ﬁberwise bilinear form, given by /a; bS ¼ ixa b ¼ ð1Þk ixb a ¼ ð1Þk /b; aS:

ð4:4Þ

Note, that /; S is skew symmetric for forms of odd degree and symmetric, but usually not deﬁnite, for forms of even degree. Its relation to the star operator is a4 * b ¼ /a; bS

on n!

and / * a; * bS ¼ /a; bS:

ð4:5Þ

Deﬁne the following graded endomorphisms on On ðMÞ: ha :¼ ðjaj nÞa; ea :¼ a4o; f a ¼ ixo a ¼ * e * a and da :¼ ð1Þjajþ1 * d * a; where jaj denotes the degree of the form a: One easily shows, see for example [Y96]: Proposition 4.1. The operators f ; d; h; d; and e make Onþ * ðMÞ into an a-module, where a is the five-dimensional Z-graded Lie algebra from Section 3. Remark 4.2. Note, that the usual grading on On ðMÞ is not the grading we considered in Section 2. More precisely Onþk ðMÞ is the eigenspace of h to the weight k: Deﬁnition 4.3. Let Ok0 ðMÞ denote the space of harmonic forms aAOk ðMÞ; da ¼ da ¼ 0: For m40 we let Okm ðMÞDOk ðMÞ denote the space of aAOk ðMÞ; such that da ¼ 0 and such that there exist aj AOk2j ðMÞ; 1pjpm; satisfying da ¼ da1 ; daj ¼ dajþ1 ; 1pjpm 1 and dam ¼ 0: For mo0 we let Okm ðMÞ denote the space of aAOk ðMÞ; such that da ¼ 0 and such that there exist aj AOkþ2j1 ðMÞ; 1pjp m; satisfying a ¼ da1 ; daj ¼ dajþ1 ; 1pjp m 1: Finally set Hmk ðMÞ :¼ Okm ðMÞ=ððimg dÞ-Okm ðMÞÞDH k ðMÞ; cf. Deﬁnition 3.1.

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Note, that H0n ðMÞ is the space of harmonic cohomology classes, those having harmonic representatives. As we have seen in Section 3, Hmn ðMÞ is a ﬁltration k of H n ðMÞ: We set H˜ km ðMÞ :¼ Hmk ðMÞ=Hm1 ðMÞ; and if H n ðMÞ is ﬁnite dimensional k k k we deﬁne b ðMÞ :¼ dim H ðMÞ; bm ðMÞ :¼ dim Hmk ðMÞ; b˜km ðMÞ :¼ dim H˜ km ðMÞ P k b ðMÞtk ; pM and the corresponding Poincare´ polynomials pM ðtÞ :¼ m ðtÞ :¼ P P k k M k k ˜ bm ðMÞt : bm ðMÞt ; p˜ m ðtÞ :¼ Since On ðMÞ is an a-module, we have a b-module structure on H n ðMÞ; e½a ¼ ½o4½a; h½a ¼ ðjaj nÞ½a; cf. Section 3. Let H n ðMÞm denote the canonic ﬁltration on the b-module H n ðMÞ from Section 2. Proof of Theorem 1.1. The ﬁrst assertion is just Proposition 3.9. For the second note, that gn : H n ðMÞ#Rn -H n ðM 0 Þ#Rn0 is a b-module homomorphism. From Proposition 2.5(2) we see that gn maps ðH n ðMÞ#Rn Þm to ðH n ðM 0 Þ#Rn0 Þm : For every bmodule V we clearly have ðV #Rn Þm ¼ Vmn ; and the proof is complete. & Proof of Theorem 1.2. The ﬁrst statement follows from Lemmas 3.2 and 3.3. Via Theorem 1.1 the remaining assertions follow from Propositions 2.6(1) and 2.6(4). & Proof of Theorem 1.3. This is an immediate consequence of Theorem 1.1 and Proposition 2.7. Proof of Theorem 1.4. On ðH n ðMÞÞn consider the b-module structure dual to the one on H n ðMÞ: Then F:H

nk

ðMÞ-ðH

nþk

n

ðMÞÞ ;

FðaÞðbÞ ¼ ð1Þ

ðnkÞðnk1Þ=2

Z a4b M

is a b-module homomorphism, and an isomorphism by ordinary Poincare´ duality. By Proposition 2.5(2) F induces isomorphisms F : H nk ðMÞm -ððH nþk ðMÞÞn Þm : From Theorem 1.1 and Lemma 2.8(1) we thus get isomorphisms F : Hmnk ðMÞ-fjAðH nþk ðMÞÞn : jjH nþk

m1

ðMÞ

¼ 0g:

So the pairing in the theorem is well deﬁned and non-degenerate. If n is even we choose a linear splitting H˜ n0 ðMÞDH0n ðMÞ and a subspace n QDH n ðMÞ; such that H n ðMÞ ¼ H1 ðMÞ"H˜ n0 ðMÞ"Q: Via this decomposition, the n n pairing H ðMÞ#H ðMÞ-R has the form 0

0 B @ 0 At

0 S Bt

1 A C BA T

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with S and T symmetric. But the signature of such a bilinear form is the signature of S; which is the signature of ð; Þ : H˜ n0 ðMÞ#H˜ n0 ðMÞ-R: & nþk ˜ nk Proof of Theorem 1.7. We have * : Onk 0 ðMÞ-O0 ðMÞ and * : H0 ðMÞnþk H˜ 0 ðMÞ: But note, that it does not induce a well-deﬁned mapping on H0n ðMÞ: Since * 2 ¼ 1 it follows from Theorem 1.4, that 0; T is non-degenerate. Since a4 * b ¼ /a; bS the statement about the symmetry follows from Eq. (4.4). &

Remark 4.4. Suppose M is a closed symplectic manifold. Since H˜ n0 ðMÞ is a g-module we can decompose it uniquely into H˜ n0 ðMÞC"lX0 Ql ; where Ql is a direct sum of g-representations with highest weight l: From ðea; bÞ ¼ ða; ebÞ we see, that if lal 0 then Ql is orthogonal to Ql 0 with respect to the pairing in Theorem 1.4. For 0pkpl and n k even we let sl ðMÞ denote the signature of the non-degenerate pairing Qnk #Qnk -R; a#b/ða; ek bÞ: It follows from l l ðea; bÞ ¼ ða; ebÞ; that this does not depend on k:3 If n is even Theorem 1.4 yields P lX0 sl ðMÞ ¼ signðMÞ: On the other hand for lal 0 ; Ql is also orthogonal to Ql 0 with respect to the pairing 0; T; for we have 0ea; bT ¼ 0a; f bT: For n k even and lpkpl we let sl ðMÞ denote the signature of 0; T : Qnk #Qnk -R: This does not depend on k since we l l 1 have 0ea; ebT ¼ 0a; febT ¼ 4ðl þ kÞðl k þ 2Þ0a; bT for all aAQnk : We claim, l that * el a ¼ ð1ÞðnlÞðnl1Þ=2 a for aAQnl l ; and thus sl ðMÞ ¼ ð1ÞðnlÞðnl1Þ=2 sl ðMÞ: ˜ nþl For that we ﬁrst show, that every ½gAQnþl l DH0 ðMÞ has a representative nþl g* AO ðMÞ with e*g ¼ 0: Indeed, by Proposition 2.6(2) we may assume 0 ¼ ½egAH0nþlþ2 ðMÞ: So there exists rAOnþlþ1 ðMÞ with eg ¼ dr and by Lemma nþl1 nþl 2.1(2) we may choose rAO * ðMÞ such that er* ¼ r: Then g* :¼ g d rAO * ðMÞ is nþl with e*g ¼ 0: Applying * we see, that every a representative of ½*g ¼ ½gAQl nl

o has a representative with f a ¼ 0: For such an a one easily veriﬁes ixa ðnlÞ! ¼ ½aAQnl l o ð1ÞðnlÞðnl1Þ=2 a and hence * el a ¼ ixol 4xa on! ¼ ixa ðnlÞ! ¼ ð1ÞðnlÞðnl1Þ=2 a: n

nl

If one considers a product M M 0 of two symplectic manifolds, then certainly a harmonic form on M and one on M 0 give rise to a harmonic form on M M 0 ; in the obvious way, cf. [I01]. But there might be additional harmonic classes on the product. More precisely, we have the following Ku¨nneth theorem for harmonic cohomology: 3 If n is odd and l ¼ 0 such a k does not exists and we set sl ðMÞ ¼ 0 in this case. Note, that if n l is odd then sl ðMÞ ¼ 0; for Qnk can only be non-trivial when n l n k mod 2: l

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Proposition 4.5. If M and M 0 are symplectic manifolds with finite-dimensional P 0 0 ðtÞ ¼ m1 þm2 ¼m p˜ M ˜M cohomology, then p˜ MM m1 ðtÞp m m2 ðtÞ: Proof. The Ku¨nneth theorem tells, that we have an isomorphism of b-modules H n ðMÞ#H n ðM 0 Þ-H n ðM M 0 Þ: Now apply Lemma 2.8(3) and Theorem 1.1. & Example 4.6. This can be used to produce an example of a closed six-dimensional symplectic manifold on which there is a non-harmonic class in dimension 3; cf. a footnote in [I01]. Indeed let N denote the Thurston nil-manifold, see [T76] or [M84]. It is a closed symplectic manifold of dimension 4 with Betti numbers b0 ðNÞ ¼ b4 ðNÞ ¼ 1; b1 ðNÞ ¼ b3 ðNÞ ¼ 3; b2 ðNÞ ¼ 4 and it has a non-harmonic class 2 3 4 3 ˜N ˜N and in dimension 3: So p˜ N 1 ðtÞ ¼ t; p 0 ðtÞ ¼ 1 þ 2t þ 4t þ 2t þ t ; p 1 ðtÞ ¼ t N ðtÞ ¼ p ˜ ðtÞ ¼ 0: Via Proposition 4.5 we get p˜ N 2 2 1

1

3 2 3 5 ðtÞ ¼ p˜ N ˜ CP p˜ NCP 1 0 ðtÞ ¼ t ð1 þ t Þ ¼ t þ t ; 1 ðtÞp

hence N CP1 has a non-harmonic class in dimension 3: Remark 4.7. The algebraic theory from Sections 2 and 3 can also be applied to forms with compact supports, or to forms with values in a ﬂat vector bundle, cf. a remark in [Y96].

5. Symplectic blow up, an example McDuff [M84] showed how to blow up a manifold X along a closed codimension 2k submanifold M; whose normal bundle admits a UðkÞ-structure. If X is symplectic and M is a symplectic submanifold this is always the case and McDuff constructed a ˜ Using this construction she produced symplectic form on the blown up manifold X: examples of closed, simply connected symplectic manifolds, which do not satisfy the ˜ Hard Lefschetz Theorem. For MDCPn we will compute p˜ X ˜M m in terms p m ; see Theorem 5.2 below. We ﬁrst recall how the blow up construction works. Following [M84] we ﬁrst show how one blows up a complex vector bundle E-M along its zero section M: Since E is assumed to be complex we have a principal UðkÞ-bundle P-M; such that P UðkÞ ˜ :¼ P UðkÞ Ck ¼ E: Since UðkÞ acts on CPk1 we have an associated bundle M CPk1 over M with ﬁber CPk1 : Moreover UðkÞ acts on the canonical line bundle L-CPk1 by bundle automorphisms, thus we also have a bundle p: * E˜ :¼ P UðkÞ L ˜ it is just the canonical over M with ﬁber L: E˜ also is a complex line bundle over M; ˜ ˜ ˜ M subbundle of M M E: Let j : E-E denote the restriction of the projection M

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˜ respectively. E-E: Finally let E0 and E˜ 0 denote the non-zero vectors in E and E; ˜ Note, that j : E0 -E0 is a diffeomorphism. We have a commutative diagram:

˜ ˜ denote the Thom class of j : E-E is called the blow up of E along M: Let aAHc2 ðEÞ 2 ˜ ˜ E-M: It restricts to a generator of H ðLÞ; where L is considered as the ﬁber of ˜ is a free H n ðMÞ˜ p* : E-M: The Leray–Hirsch theorem now implies, that H n ðEÞ module with basis f1; a; y; ak1 g; cf. [M84]. Note, that the aj have compact supports for jX1: If M is a closed submanifold of X whose normal bundle E admits a UðkÞ˜ structure, one identiﬁes a tubular neighborhood V of M with E and replaces it by E: ˜ see [M84]. This is McDuff’s deﬁnition of the blow up X; ˜ is injective.4 Since aj It follows from Poincare´ duality, that jn : H n ðX Þ-H n ðXÞ have support in V˜ :¼ j1 ðV Þ we have a well deﬁned and injective ˜ H n ðMÞ#R½a; a2 ; y; ak1 -H n ðXÞ;

b#aj /ðp* n bÞ4aj ;

˜ where p* : V-M: In [M84] it is shown, that the image of these two injective mappings n ˜ span H ðXÞ additively, that is ˜ ¼ ðjn ðH n ðX ÞÞÞ"ðp* n ðH n ðMÞÞ#R½a; a2 ; y; ak1 Þ: H n ðXÞ

ð5:6Þ

If ðM; sÞ is a symplectic submanifold of ðX ; oÞ McDuff constructed a symplectic ˜ V˜ and such that ½oj ˜ ˜ such that jn o ¼ o * on X\ * V˜ ¼ p* n ½s þ eaAH 2 ðVÞ; * on X; form o for some e40: Here, and from now on, we assume kX2; but note, that for k ¼ 1 one has X˜ ¼ X : ˜ Note, that this is * ¼ jn ½o þ eaAH 2 ðXÞ: Lemma 5.1. With these notations we have ½o * with respect to (5.6). Moreover we have a short exact sequence the decomposition of ½o of b-modules n ˜ H n ðMÞ#W -H n ðXÞ-H ðX Þ;

where W is the b-module with base e ðeaÞj :¼ ðeaÞjþ1 ; h ðeaÞj :¼ ð2j kÞðeaÞj :

fðeaÞ; y; ðeaÞk1 g

ð5:7Þ and

4

action

Indeed, suppose conversely 0aaAH l ðX Þ and jn a ¼ 0: Using Poincare´ duality we ﬁnd ˜ ˜ ˜ is @ XÞ bAHcdim X l ðX ; @X Þ; such that 0aa4bAHcdim X ðX ; @X Þ: Clearly jn : Hcdim X ðX ; @X Þ-Hcdim X ðX; injective, hence jn a4jn ba0 and thus jn aa0; a contradiction. So jn has to be injective.

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% Proof. Let V1 +V+V be a slightly larger tubular neighborhood of M; set V˜ 1 :¼ 1 j ðV1 Þ and consider the following piece of the Mayer–Vietoris sequence: @ 1 ˜ 2 ˜ ˜ ˜ VÞ"H ˜ ˜ ˜ ðV1 Þ-H 1 ðV˜ 1 \VÞ H 2 ðXÞ-H ðX\VÞ"H 2 ðV˜ 1 Þ: H 1 ðX\

ð5:8Þ

˜ V˜ and since ½oj ˜ the * ¼ jn o on X\ * V˜ ¼ p* n ½s þ ea ¼ ðjn ½oÞjV˜ þ eaAH 2 ðVÞ Since o n * j ½o eaAimg @: Below we will show, that exactness of (5.8) yields ½o ˜ is onto, then @ ¼ 0 and ½o * ¼ jn ½o þ ea: For this it sufﬁces H 1 ðV˜ 1 Þ-H 1 ðV˜ 1 \VÞ 1 ˜ ˜ to show, that H 1 ðEÞ-H 1 ðE0 Þ is onto, for then H 1 ðEÞ-H ðE0 Þ is also onto, which 1 ˜ 1 ˜ ˜ is equivalent to the surjectivity of H ðV1 Þ-H ðV1 \VÞ: For that ﬁrst note, that the Thom isomorphism of the normal bundle E-M gives H 2 ðE; E0 Þ ¼ Hc2 ðEÞDH 22k ðMÞ ¼ 0; for we have 2 2ko0: Thus the Gysin sequence ?-H 1 ðEÞ-H 1 ðE0 Þ-H 2 ðE; E0 Þ ¼ 0 shows that H 1 ðEÞ-H 1 ðE0 Þ is onto. This proves the ﬁrst assertion. The second immediately follows from n * ½o4ðj a þ ðp* n bÞ4ðeaÞj Þ

¼ ðjn ½o þ eaÞ4ðjn a þ ðp* n bÞ4ðeaÞj Þ ¼ ðjn ð½o4aÞÞ þ ððp* n in aÞ4ðeaÞ þ p* n ð½s4bÞ4ðeaÞj þ ðp* n bÞ4ðeaÞjþ1 Þ; where i : M-X denotes the inclusion.

&

Theorem 5.2. Suppose ðM; sÞ is a closed codimension 2k symplectic submanifold ˜ oÞ * denote the blow up of CPn along M: Then for all mAZ of CPn ; and let ðX; we have ˜

n

2 4 2k2 p˜ X ˜ CP ˜M Þ: m ðtÞ þ p m ðtÞ ¼ p m ðtÞðt þ t þ ? þ t n

n

2 2n Here p˜ CP and p˜ CP m ðtÞ ¼ 0; for ma0: 0 ðtÞ ¼ 1 þ t þ ? þ t

Proof. In the case X ¼ CPn sequence (5.7) certainly splits, cf. the proof of Lemma 2.3, and thus via Lemma 2.8(2) ˜

n

p˜ X ˜ CP ˜H m þp m ¼ p m

n

ðMÞ#W

:

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2 4 2k2 Since p˜ W ˜W ; Lemma 2.8(3) implies m ¼ 0 for ma0 and p 0 ðtÞ ¼ t þ t þ ? þ t X n ðMÞ#W 2 4 2k2 ðtÞ ¼ p˜ M ˜W ˜M Þ; p˜ H m m1 ðtÞp m2 ðtÞ ¼ p m ðtÞðt þ t þ ? þ t m1 þm2 ¼m

and the theorem is proved. & Remark 5.3. Summing over m40; Theorem 5.2 especially gives ˜

˜

M M 2 4 2k2 ðpX pX Þ: 0 ÞðtÞ ¼ ðp p0 ÞðtÞðt þ t þ ? þ t

ð5:9Þ

Note, that p p0 measures how many classes are not harmonic. Thus X˜ satisﬁes the Hard Lefschetz Theorem if and only if M does. Moreover (5.9) tells how many ˜ are not harmonic. As a special case we recover cohomology classes in H n ðXÞ McDuff’s example from [M84], where she assumed that all Chern classes of M vanish, and proved that X˜ does not satisfy the Hard Lefschetz Theorem, provided M does not. ˜

Remark 5.4. The same method permits to compute p˜ X m whenever sequence (5.7) splits. For instance this is always the case if M is a point in an arbitrary closed symplectic manifold X :

References [B88] [I01] [M95] [M84] [T76] [Y96]

J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988) 93–114. R. Iba´n˜ez, Yu. Rudyak, A. Tralle, L. Ugarte, On symplectically harmonic forms on sixdimensional nil-manifolds, Comment. Math. Helv. 76 (2001) 89–109. O. Mathieu, Harmonic cohomology classes of symplectic manifolds, Comment. Math. Helv. 70 (1995) 1–9. D. McDuff, Examples of simply-connected symplectic non-Ka¨hlerian manifolds, J. Differential Geom. 20 (1984) 267–277. W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467–468. D. Yan, Hodge structure on symplectic manifolds, Adv. in Math. 120 (1996) 143–154.

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