70 Group extensions and Kähler groups
S1-bundle. Conversely, every principalS1-bundle over a classifying spaceBQgives rise to a group extension by looking at the long exact sequence of homotopy groups. More-over, the total space of such anS1-bundle is aspherical, hence a classifying space for its fundamental group. Therefore central extensionsΓof a groupQbyZare classified by their Euler class of the associated principalS1-bundle inH2pQ;Zq. We will call this the Euler class of the central extensionand denote it byepΓq.
Remark4.6. This is not the usual definition of the characteristic class of a central ex-tension. In general central extensions
0ÝÑCÝÝÑi ΓÝÑÝp QÝÑ0
are classified by their characteristic class in H2pQ;Cq. Since we will not need this classification in full generality, we use our simplified definition.
Next we give a construction of the classifying space Bπ1pMqfor a manifold M. In fact, we will also construct a classifying mapM ÝÑBπ1pMq. LetΓbe the fundamental group of a manifold M. Then a classifying space BΓ for Γ can be constructed in the following way. We attach cells of dimension 3 toMalong generators ofπ2pMqin order to get a space M2 withπ2pM2q “ 0. Subsequently, we attach 4-cells in order to get a spaceM3such thatπ3pM3q “0 and so on. Since we only attached cells of dimension 3 or higher, the result is an aspherical spaceM8 which has the same fundamental group asM. HenceM8 “BΓ. Thus we have a natural inclusion
ι: MÝÑ BΓ. (4.3)
By definitionιinduces an isomorphism
ι˚: H1pBΓq ÝÑH1pMq (4.4)
and a injection
ι˚: H2pBΓq ÝÑH2pMq. (4.5) In particular it follows that b1pΓq “ b1pMqfor any manifold Mwithπ1pMq “Γ.
Now suppose X is an orbifold. Then we can replicate the above construction on the orbifold classifying spaceBX. We obtain a map ι: BX ÝÑ Bπorb1 pXqsuch that the homomorphisms (4.4) and (4.5) satisfy the same properties when replacing H˚pMqby Horb˚ pXq.
4.2 Kähler groups 71
interestingly, do not hold in Sasakian setting. If no reference is provided, the proofs of the results in this section can be found in [2].
Definition 4.7. AKähler group is the fundamental group of a compact Kähler man-ifold. Analogously, the fundamental group of a smooth projective variety over C is a projective group. Denote byK, respectivelyP, the set of Kähler groups, resp. projec-tive groups. We defineK2n, resp. P2n, to be the set of fundamental groups of compact Kähler manifolds, resp. projective manifolds, of real dimension 2n.
LetX be a 2n-dimensional Kähler manifold withπ1pXq “ Γ. Given a finite index subgroup Γ1 Ă Γ, we can lift the Kähler structure to the compact covering space X1 associated toΓ1. This simple observation immediately yields the following property of Kähler groups.
Proposition 4.8. The setK2n is closed under taking finite index subgroups.
Clearly the product of two Kähler, respectively projective, manifolds is again Käh-ler, resp. projective. Thus the sets P and K are closed under taking direct products.
Moreover, taking cartesian products withCP1increases the dimension in which a Käh-ler or projective group can be realized. We collect these properties in a proposition for future reference.
Proposition 4.9. The classes of Kähler and projective groups enjoy the following prop-erties.
i) The setsPandK are closed under taking direct products.
ii) There are inclusionsK2n ĂK2n`2 andP2nĂP2n`2for all n.
Under a natural dimension restriction, one can prove the converse of ii) in Proposi-tion 4.9 in the projective setting. This is due to the following version of the Lefschetz Hyperplane Theorem proven by Bott.
Theorem 4.10 (Lefschetz Hyperplane Theorem [13]). Let X Ă CPN be a projective variety of complex dimension n. Consider a hyperplane section Y ÝÑ X given by a hyperplane transverse to X. Then the induced map on homotopy groups
πipYq ÝÑπipXq is an isomorphism for all iăn.
Corollary 4.11. Every projective group is realizable in real dimension4. In particular there are bijectionsP2i “P2jfor all i, jě2.
Remark4.12. Whether the same result holds for Kähler groups is still an open problem.
Remark 4.13. It follows from Kodaira’s classification of complex surfaces that the set P4 coincides with the set K4. Therefore, every projective group is the fundamental group of a 4-dimensional Kähler manifold, i.e. P“P4“K4.
72 Group extensions and Kähler groups
The next properties of Kähler groups that we will review are consequences of Hodge theory. In particular, the following classical result have important implications for Käh-ler groups.
Theorem 4.14(Hard Lefschetz Theorem). LetpX, ωqbe a compact Kähler manifold of complex dimension n. Then taking the wedge product with the k-th power of the Kähler class induces an isomorphism
Lk: HdRn´kpXq ÝÑHdRn`kpXq rαs ÞÑ rα^ωks for all kďn.
Moreover, Hodge theory implies that the first Betti number b1pXqof a Kähler man-ifold is even. This follows from Hodge decomposition and the fact that complex con-jugation yields an isomorphism in Dolbeault cohomology. Then the construction of the inclusion (4.3) shows that b1pΓqis even. Therefore we have the following:
Proposition 4.15. The first Betti number of a Kähler group is even.
Proposition 4.15, in turn, implies the following Corollary 4.16. Free groups are not Kähler.
Proof. Let Fn be the free group onn generators. Then b1pFnq “ n. This rules out all free groups on an odd number of generators. In all other cases Fn admits finite index subgroups which are isomorphic to FN for N odd. Therefore the claim follows from
Proposition 4.8.
This is not the only application of Hodge theory to the study of Kähler groups. In particular, Theorem 4.14 has a non-trivial consequence which we present now. LetΓbe the fundamental group of a compact Kähler manifoldXof real dimension 2n. Consider the isomorphism Ln´1: HdR1 pXq ÝÑ H2n´1dR pXq. Combining this with Poincaré duality we get a non-degenerate bilinear pairing
HdR1 pXqˆH1dRpXq ÝÑR
pα,βq ÞÝÑ xαYβYωn´1,rXsy.
which factorizes through the cup product
HdR1 pXq ˆHdR1 pXq ÝÑHdR2 pXq.
We have seen in the previous section that there exists a mapι: X ÝÑ BΓ whereΓ “ π1pXq. This map induces an isomorphism
ι˚: H1pBΓq ÝÑH1pXq
4.2 Kähler groups 73
and a injection
ι˚: H2pBΓq ÝÑH2pXq.
This is explained in the discussion before (4.4) and (4.5). Combining this with the previous discussion we get the following:
Proposition 4.17. Let Γ be a Kähler group. Then there is a non-degenerate skew-symmetric bilinear product
H1pΓq ˆH1pΓq ÝÑR (4.6)
which factorizes through the cup product
H1pΓq ˆH1pΓqÝÝÑY H2pΓq.
Proposition 4.18. Letπ: M ÝÑ Xbe the principal orbibundle associated to a quasi-regular Sasaki structure. Then there is a non-degenerate skew-symmetric bilinear prod-uct
H1pπorb1 pXq;Rq ˆH1pπorb1 pXq;Rq ÝÑR (4.7) which factorizes through the cup product
H1pπorb1 pXq;Rq ˆH1pπorb1 pXq;RqÝÝÑY H2pπorb1 pXq;Rq.
Proof. Theorem 3.78 is the analogue of the Hard Leftschetz Theorem in basic cohomol-ogy of a Sasaki structure. In the quasi-regular case the basic cohomolcohomol-ogy ringH˚BpF;Rq coincides with the orbifold cohomology ring H˚orbpX;Rq. Therefore, the claim follows from the fact that the homomorphisms (4.4) and (4.5) are defined also in the orbifold
case.
Example4.19 (The Heisenberg groupH3). LetT2 “ S1ˆS1 be the two dimensional torus. Consider the classesα1, α2 P H1pT2;Zqgiven by the generators of the cohomol-ogy of the two factors. Denote byβ “ α1Yα2 the generator of H2pT2;Zq. Now let M be the principalS1-bundle on the torus T2 with Euler class β. The 3-dimensional Heisenberg group H3 is the fundamental group π1pMq. Clearly M is aspherical, thus M “ BH3. Since H2pT2;Zq – Z and the Euler class is a generator, it follows from the Gysin sequence of the principalS1-bundle thatH2pMq “ 0. It is then evident that the cup product of classes inH1pMqvanishes. Hence a non-degenerate skew-symmetric bilinear product onH1pH3qcannot factorize through the cup product. We conclude that H3is not a Kähler group.
Example 4.20 (Higher rank Heisenberg groups). Now let T2n be the 2n-dimensional torus and letαibe the generator of the integral cohomology of thei-th factor. Denote by βthe classřn
i“1α2i´1Y α2i. Then we define the 2n`1-dimensional Heisenberg group H2n`1to be the fundamental group of the principalS1-bundle determined byβ. Carlson
74 Group extensions and Kähler groups
and Toledo [27] proved that H5 and H7 are not Kähler groups while Campana [20]
proved thatH2n`1 is Kähler forně4.
Johnson and Rees [66] used Proposition 4.17 to show that the free product of groups with non-trivial finite quotients is not a Kähler group. In fact, they proved a much more general statement. The proof we give here is a simplified version of the original proof, relying on Lemma 4.21.
Lemma 4.21([78]). LetΓ1andΓ2be two groups. Assume fi: Γi ÝÑQiis a non-trivial quotient with kernel Ki and|Qi| “ mi ă 8 for i “ 1,2. Then the free product Γ1˚Γ2
admits a finite index subgroup with odd first Betti number.
Proof. Consider the following homomorphism Γ1˚Γ2
πab
Ý
ÝÑΓ1ˆΓ2 f1ˆf2
ÝÝÝÑQ1ˆQ2 .
By the Kurosh subgroup theorem, the kernel of the above homomorphism has the form Fm˚KwhereFmis the free group onm“ pm1´1qpm2´1qgenerators andK “K1˚K2. Now let f: Fm ÝÑ Qbe a finite quotient with|Q| “ d. Extend f trivially onK to get a homomorphism ¯f: Fm˚K ÝÑQ. Then the kernel of ¯f has the formFn˚K˚ ¨ ¨ ¨ ˚K wheren “ 1`dpm´1qand K appears d many times. Thus, kerpf¯qis a finite index subgroup inΓ1˚Γ2and
b1pkerpf¯qq “ n`db1pKq “1`dpm´1`b1pKqq.
By pickingd“2cwe get a finite index subgroup ofΓ1˚Γ2with odd first Betti number.
Theorem 4.22 ([66]). Let Γ1 andΓ2 be groups admitting a non-trivial finite quotient.
Then the group
Γ“ pΓ1˚Γ2q ˆH
is not Kähler for any group H. In particularΓ1˚Γ2 is not a Kähler group.
Proof. Suppose that the first Betti number b1pHqof the groupHis even. By Lemma 4.21, there exists a finite index subgroup∆ĂΓ1˚Γ2with b1p∆qodd. Hence, the group∆ˆH is a finite index subgroup ofΓwith odd first Betti number. ThusΓcannot be Kähler by Proposition 4.8 and Proposition 4.15.
If, instead, the first Betti number b1pHqis odd, then b1pΓ1˚Γ2q ą 0. Thus we can assume that the first Betti number ofΓ1is positive. The proof of Lemma 4.21 provides a finite index subgroup ofΓ1ˆΓ2 of the form Fn˚G whereFn is the free group onn generators. Moreover, the rank of Fn can be chosen to be arbitrarily large. Since the class of Kähler groups is closed under taking finite index subgroups, we can assume Γ“ pFn˚Gq ˆH withnąb1pHq. Moreover, the bilinear product
H1pΓq ˆH1pΓq ÝÑR