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A Note on Compact Solvmanifolds with K ¨Ahler Structures

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A Note on Compact Solvmanifolds with K ¨Ahler Structures View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Osaka City University Repository Hasegawa, K. Osaka J. Math. 43 (2006), 131–135 A NOTE ON COMPACT SOLVMANIFOLDS WITH KAHLER¨ STRUCTURES KEIZO HASEGAWA (Received July 26, 2004, revised May 17, 2005) Abstract In this note we show that a compact solvmanifold admits a Kahler¨ structure if and only if it is a finite quotient of a complex torus which has a structure of a com- plex torus bundle over a complex torus. We can show in particular that a compact solvmanifold of completely solvable type has a Kahler¨ structure if and only if it is a complex torus, which is known as the Benson-Gordon’s conjecture. 1. Introduction We know that the existence of Kahler¨ structure on a compact complex manifold imposes certain homological or even homotopical restrictions on its underlining topo- logical manifold. Hodge theory is of central importance in this line. There have been recently certain extensions and progresses in this area of research. Among them is the field of Kahler¨ groups, in which the main subject to study is the fundamental group of a compact Kahler¨ manifold (see [1]). Once there was a conjecture that a non-abelian, finitely generated and torsion-free nilpotent group (which is the fundamental group of a nilmanifold) can not be a Kahler¨ group, which is a generalized assertion of the result [9, 14] that a non-toral nilmanifold admits no Kahler¨ structures. A counter-example to this conjecture was given by Campana [12]. Later a detailed study of solvable Kahler¨ groups was done by Arapura and Nori [3]; they showed in particular that a solvable Kahler¨ group must be almost nilpotent, that is, it has a nilpotent subgroup of finite index. On the other hand, the author stated in the paper [15] a general conjecture on compact Kahlerian¨ solvmanifolds: a compact solvmanifold admits a Kahler¨ structure if and only if it is a finite quotient of a complex torus which is also a complex torus bundle over a complex torus; and showed under some restriction that the conjecture is valid. In this note we will see that the above conjecture can be proved without any re- striction, based on the result (mentioned above) by Arapura and Nori [3], and applying the argument being used in the proof of the main theorem on the author’s paper [15]. We also see that the Benson-Gordon’s conjecture on Kahler¨ structures on a class of compact solvmanifolds [10] (so-called solvmanifolds of completely solvable type) can 2000 Mathematics Subject Classification. Primary 32Q15, 32M10; Secondary 53C30, 53D05. 132 K. HASEGAWA be proved as a special case of our main result. In this note we mean by a solvmanifold (nilmanifold) a compact homogeneous space of solvable Lie group, that is, a compact differentiable manifold on which a connected solvable (nilpotent) Lie group acts transitively. We can assume, by tak- Å ing the universal covering group of , that a solvmanifold is of the form , where is a simply connected solvable Lie group and is a closed subgroup of (which contains no non-trivial connectede normal subgroup of ). It should bee noted that unlesse Å is a nilmanifold, a closed subgroup may not be a discrete subgroupe (a lattice) of . However, it is known (due to Auslander [4])e that a solvmanifold in ¼ general has a solvmanifold ¼ with discrete isotropy subgroup as a finite (normal) covering. We recall some terminologiese which we use in this note. A solvmanifold Å = ¼ ¼ , where is a discrete subgroup of a simply connected solvable Lie group , is of completely solvable type, if the adjoint representation of the Lie algebra g of has only real eigenvalues; and of rigid type (or of type (R)), in the sense of Auslander [5], if the adjoint representation of g has only pure imaginary (including 0) eigenvalues. It is clear that Å is both of completely solvable and of rigid type if and only if g is nilpotent, that is, Å is a nilmanifold. We can see in the proof of main theorem that a Kahlerian¨ solvmanifold is of rigid type, and not of completely solvable type unless it is a complex torus. This gives a proof of the Benson-Gordon’s conjecture as a special case of the theorem. We state now our main theorem in the most general form: Main Theorem. A compact solvmanifold admits a Kahler¨ structure if and only if it is a finite quotient of a complex torus which has a structure of a complex torus bun- dle over a complex torus. In particular, a compact solvmanifold of completely solvable type has a Kahler¨ structure if and only if it is a complex torus. 2. Proof of main theorem Ñ Let Å be a compact solvmanifold of dimension 2 which admits a Kahler¨ struc- ture. We will first show that Å is a finite quotient of a complex torus. We can as- ¼ ¼ sume, taking a finite covering if necessary, that Å is of the form , where is a lattice of a simply connected solvable Lie group . By the result of Arapura and Nori, Å ¼ we know that the fundamental group ¼ of is almost nilpotent, that is, contains a ½ nilpotent subgroup ½ of finite index. Then we can find a normal nilpotent subgroup ′ ½ of ¼ , which is commensurable with , and thus defines another lattice of . There- ½ Å fore Å ′ = ′ is a finite normal covering of , which is a solvmanifold with the fundamental group ½′, a finitely generated torsion-free nilpotent group. According to the well-known theorem of Mostow [18], Å ′ is diffeomorphic to a nilmanifold. Since Å Å ′ has a canonical Kahler¨ structure induced from , it follows from the result on Å Kahlerian¨ nilmanifolds [9, 14] that Å ′ must be a complex torus. Hence is a finite COMPACT SOLVMANIFOLDS WITH KAHLER¨ STRUCTURES 133 quotient of a complex torus. We will show that Å has a structure of a complex torus bundle over a complex torus. We follow the argument in the proof of the main theorem on the paper [15] to Å which we refer for full detail. Let ¼ be the fundamental group of . Then we can Æ express ¼ as the extension of a finitely generated and torsion-free nilpotent group of rank 2Ð by the free abelian group of rank 2 , where we can assume 2 is the first Å Ñ Ð Betti number 1 of (we have 2 1 in general) [7] and = + : ≤ 2 ¼ 0 Æ Z 0 → → → → 2Ñ Ì ¼ Since Å is also a finite quotient of a torus , contains a maximal normal free Ñ ¼ Æ abelian subgroup ½ of rank 2 with finite index in . We can see that must be Æ ½ Æ free abelian of rank 2Ð . In fact, since is a abelian subgroup of with finite in- ∩˜ Æ Æ dex, it follows that the real completion Æ of is abelian, and thus is also abelian. Therefore we have the following 2Ð 2 0 Z ¼ Z 0 → → → → 2Ð ½ × × × À × × × where = Z 1Z 2Z 2 Z, and = Z 1Z Z 2Z Z 2 Z (some × × ×···× × ×···× Ñ × ¼ Å ¼ of Z Z may be trivial) is the holonomy group of . We have now that = C , À À where ¼ is a Bieberbach group with holomomy group . Since the action of on Ð Ð 2 Å C Z is holomorphic, we see that is a holomorphic fiber bundle over the complex Ð Ð 2 2 torus C Z with fiber the complex torus C Z . Conversely, let Å be a finite quotient of a complex torus which is also a complex Å torus bundle over a complex torus. The fundamental group ¼ of is a Bieberbach group which is expressed as the extension of a free abelian group Z2Ð by another free Ð abelian group Z2 . As observed before, since the action of Z2 on Z2 is actually the 2Ð ¼ À action of the finite abelian holonomy group À of on Z , and the action of on 2 the fiber is holomorphic, we may assume that the action of Z is in U(Ð ) and thus Ð 2 2 extendable to the action of R on R , defining a structure of solvmanifold on Å of ¼ the form ¼ , where is a lattice of a simply connected solvable Lie group . In Ð Ð Ð 2Ð 2 2 fact, considering R as C , we have = C ⋊ R with the action : R Aut(C ) → defined by Ø Ø Ø √ 1 √ 1 √ 1 1 2 Ð Þ Ø Þ Þ Þ Þ Þ Ð Ð ( )(( 1 2 )) = − 1 − 2 − 2 √ 1 Ø Ø × where = ( : the -th unit vector in R ), and − is the primitive -th root Ð of unity, = 1 2 , = 1 2 2 . As seen in the first part of the proof, we can always take 2 as the first Betti number of Å , and makes the fibration the Albanese map into the Albanese torus. Let g be the Lie algebra of ; then we can express g as a vector space over R having Ð Ð a basis 1 2 2Ð 2 +1 2 +2 for which the bracket multiplications are { } 134 K. HASEGAWA defined by the following: Ð [ 2Ð + 2 1]= 2 [ 2 + 2 ]= 2 1 − − − × Ð for all with = 1, = 1 2 2 , = 1 2 , and all other brackets vanish. It 6 is clear that g is of rigid type, and is of completely solvable type if and only if g is abelian, that is, Å is a complex torus. This completes the proof of the theorem. REMARK. 1. A flat solvmanifold is a solvmanifold with flat Riemannian met- ric (which may not be invariant by the Lie group action).
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