EFFECTIVE TRANSITIVE ACTIONS OF THE UNITARY GROUP ON QUOTIENTS OF HOPF MANIFOLDS

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1 EFFECTIVE TRANSITIVE ACTIONS OF THE UNITARY GROUP ON QUOTIENTS OF HOPF MANIFOLDS ALEXANDER ISAEV arxiv: v2 [math.cv] 25 Sep 2016 Abstract. I artice [IKru] we showed that every coected compex maifod of dimesio ě 2 that admits a effective trasitive actio by hoomorphic trasformatios of the uitary group U is bihoomorphic to the quotiet of a Hopf maifod by the actio of Z m for some iteger m satisfyig p,mq 1. I this ote, we compemet the above resut with a expicit descriptio of a effective trasitive actios of U o such quotiets, which provides a aswer to a 10-year od questio. 1. Itroductio I [IKru] we cassified a coected compex maifods of dimesio ě 2 that admit a effective actio of the uitary group U by hoomorphic trasformatios. I particuar, i Theorem 4.5 of [IKru] we showed that if the actio of U is trasitive, the the maifod i questio is bihoomorphic to a certai quotiet of a Hopf maifod. Reca that for a ozero compex umber d with d 1 the correspodig -dimesioa Hopf maifod, say Md, is costructed by idetifyig every poit z P C zt0u with the poit d z (hece with each of the poits d k z for k P Z). Ceary, Md is compact. Further, ettig rzs P M d be the equivaece cass of z P C zt0u, for m P N we deote by Md {Z m the compact compex maifod obtaied from Md 2πi by idetifyig rzs with re m zs (hece with each of the poits re 2πiK m zs for K 0,...,m1). Theorem 4.5 of [IKru] the asserts that if the group U acts o a coected compex maifod X of dimesio ě 2 effectivey ad trasitivey by hoomorphic trasformatios, the X is bihoomorphic to Md {Z m for some m P N satisfyig p,mq 1. Coversey, as the foowig exampe shows (cf. [IKru, Exampe 4.1]), every maifod Md {Z m with p,mq 1 admits a effective trasitive actio of U. Exampe 1.1. First, we defie a actio ofu o Md. Let λ be a compex umber such that e 2πpλiq d. Write ay eemet A P U as A e it B, where t P R ad B P SU, ad set (1.1) A rzs : re λt Bzs. To verify that this actio is we-defied, cosider ay other represetatio of A as above, i.e., A e ipt`2πk `2πq pe2πik Bq for some P Z, k P t0,..., 1u. The formua (1.1) yieds Arzs re λpt`2πk `2πq e2πik Bzs rd k` e λt Bzs re λt Bzs as required. Next, actio (1.1) is trasitive ad effective. Ideed, to verify its effectiveess, et pe it Bqrzs rzs for some t P R, B P SU ad a z P C zt0u. The for Mathematics Subject Cassificatio: 32M10, 32M05 Keywords: Hopf maifods, trasitive group actios.

2 2 ISAEV some k P t0,..., 1u we have B e 2πik id, ad there exists P Z such that e λt e 2πik d. Therefore, usig the defiitio of λ we obtai t 2π, e 2πik e2πi, which impies e it B id as caimed. Now, fix m P N with p,mq 1, cosider the quotiet Md {Z m ad for rzs P Md deote by trzsu P Md {Z m the equivaece cass of rzs. We defie a actio of U o Md {Z m by the formua Atrzsu : tarzsu for A P U. This actio is ceary trasitive. To verify its effectiveess, et pe it Bqtrzsu trzsu for some t P R, B P SU ad a z P C zt0u. The for some k P t0,...,1uwe haveb e 2πik id, ad there exist P Z, K P t0,...,m1u such that e λt e 2πik e 2πiK m d. Usig the defiitio of λ we the see (1.2) t 2π, e 2πik e2πi `2πiK m. Sice p,mq 1, thesecodidetityi(1.2)yiedsk 0, adweobtaie it B id as required. Sice the time artice[ikru] appeared i prit, we have bee asked o umerous occasios whether there exists a reasoabe compete descriptio of a effective trasitive actios of U by hoomorphic trasformatios o M d {Z m where oe assumes p,mq 1. I this ote we, somewhat beatedy, give a positive aswer to this questio, which is ow more tha te years od. Our resuts are summarized as foows. First, we obtai: THEOREM 1.2. If for some m P N the quotiet Md {Z m admits a effective actio of U by hoomorphic trasformatios, the p,mq 1. Further, every effective trasitive actio of U o Md {Z m by hoomorphic trasformatios has either the form (1.3) Atrzsu! e ip1`pp` q m qqt d rt CBC1 2π z (1.4) Atrzsu! e ip1`pp` q m qqt d rt CBC1 2π z, where A P U is represeted as A e it B with t P R ad B P SU, p,q,r P Z, r 0, C P GL pcq, ad for ay µ P R we set d µ : d µ e iµargd. Remark 1.3. For 2 every actio of the form (1.4) is i fact a actio of the form (1.3) sice compex cojugatio B ÞÑ B is a ier automorphism of SU 2 ; it ˆ0 i coicides with the cojugatio by the eemet. i 0 Remark 1.4. I the statemet of Theorem 1.2 we chose to defie d µ for µ P R as d µ : d µ e iµargd, where arg is the ordiary argumet fuctio takig vaues i the semiope iterva r0,2πq. Ay other possibe defiitio of d µ differs from ours by a factor of the form e 2πiµL, where L P Z is idepedet of µ. Notice that for ay such aterative defiitio of d µ formuas (1.3), (1.4) remai vaid with p repaced by p Lr. Observe that i Theorem 1.2 we do ot caim that either of formuas (1.3), (1.4) aways defies a effective actio despite the fact that these formuas were obtaied uder the effectiveess assumptio. Necessary ad sufficiet coditios o the itegers p, q, r that guaratee effectiveess are give beow:

3 ACTIONS OF THE UNITARY GROUP ON QUOTIENTS OF HOPF MANIFOLDS 3 Propositio 1.5. We have: (i) actio (1.3) is effective if ad oy if there do ot exist P Z, K P t0,...,m1u such that the foowig coditios are satisfied: paq r 1 ` p ` q m K m P Z, pbq p ` q K r m m R Z; (ii) actio (1.4) is effective if ad oy if there do ot exist P Z, K P t0,...,m1u such that (b) ad the foowig coditio are satisfied: pcq 1 ` p ` q K r m m P Z. Fiay, as oe ca see from the foowig coroary, the formuas from Theorem 1.2 ad the coditios from Propositio 1.5 sigificaty simpify i the case of Hopf maifods (i.e., whe m 1): Coroary 1.6. Every effective trasitive actio of U o Md trasformatios has either the form ı (1.5) Arzs e ip1`pqt d rt CBC1 2π z (1.6) Arzs ı e ip1`pqt d rt CBC1 2π z, by hoomorphic where A P U is represeted as A e it B with t P R ad B P SU, p,r P Z, r 0, C P GL pcq. Furthermore, actio (1.5) is effective if ad oy if there does ot exist P Z such that r divides p1 `pq but does ot divide p. Actio (1.6) is effective if ad oy if there does ot exist P Z such that r divides p1 `pq but does ot divide p. It is a easy exercise to show that actio (1.1) from Exampe 1.1 ca be writte i the form (1.5) with C id, some p P Z ad r The proofs Proof of Theorem 1.2. Reca that for ay coected compact compex maifod X the group AutpXq of its hoomorphic automorphisms is a compex Lie group i the compact-ope topoogy (see [BM]). As oted i the proof of Theorem 4.5 i [IKru], the compex Lie group AutpMd {Z mq is aturay isomorphic to the compex Lie group G : pgl pcq{hq{z m, where H : td k id, k P Zu Ă GL pcq ad Z m is idetified with the subgroup of GL pcq{h that cosists of a eemets of the form e 2πiK m H, K P t0,...,m 1u. We wi ow fid the maxima compact subgroups of G. First, cosider G : td t id,t P Ru Ă GL pcq. The the subgroup K : G{H Ă GL pcq{h is isomorphic to S 1 (with the isomorphism give by pd t idqh ÞÑ e 2πit ). We view K as a subgroup of G by way of the embeddig g ÞÑ gz m, g P K. Further, takig ito accout the atura embeddig of U i GL pcq{h by the map g ÞÑ gh, g P U, we cocude that pu {Z m q K is a maxima compact subgroup of G, where we otice that pu {Z m qxk teu. Sice K ies i the ceter of G, ay other maxima compact subgroup of G has the form s 0 pu {Z m qs1 0 K for some s 0 P G. Suppose ow that we are give a effective actio of U o M d {Z m by hoomorphic trasformatios. Ceary, the actio iduces a embeddig τ : U Ñ G. Sice τpuq is a compact subgroup of G, we have τpuq Ă s 0 pu {Z m qs1 0 K for some s 0 P G. Cosider the restrictio of τ to SU. Sice there does ot exist a otrivia

4 4 ISAEV homomorphism of SU to S 1 (which foows, e.g., from [IKra, Lemma 2.1]), we have τpsu q Ă s 0 pu {Z qs1 m 0. As the actio is effective, τpsu q is isomorphic to SU. Ceary, s 0 pu {Z qs1 m 0 cotais a subgroup isomorphic to SU if ad oy if p,mq 1, i which case τpsu q s 0 SU s1 0, where i the right-had side SU is embedded i G i the stadard way. Hece, there exists a automorphism γ of SU ad D P GL pcq such that Btrzsu trdγpbqd1 zsu for a z P C zt0u ad B P SU. Next, every automorphism of SU has either the form (2.1) B ÞÑ h 0 Bh1 0 (2.2) B ÞÑ h 0 Bh1 0 for some h 0 P SU (cf. Remark 1.3). If γ has the form (2.1), the there exists C P GL pcq such that (2.3) Btrzsu trcbc1 zsu for a z P C zt0u ad B P SU. If γ has the form (2.2), the there exists C P GL pcq such that (2.4) Btrzsu trcbc1 zsu for a z P C zt0u ad B P SU. Suppose first that SU acts as i (2.3). Restrict τ to the ceter Z of U. Composig τ with the projectios of the group s 0 pu {Z qs1 m 0 K to its first ad secod factors, we see that there exist homomorphisms τ 1 : Z Ñ s 0 pz{z qs1 m 0 Z{Z m ad τ 2 : Z Ñ K such that τpgq τ 1 pgq τ 2 pgq for a g P Z. Ceary, we have τ 1 pe it idq ppe iσt idqhqz m for some σ P R. Further, there exists µ P R such that τ 2 pe it idq pd µt HqZ m. Sice τ 2 pe it idq τ 2 pe ipt`2πq idq, the umber µ has to be of the form µ L 2π for some L P Z. Further, sice by (2.3) the map τ 2 is trivia o the ceter of SU, we obtai that L r for some r P Z. Aso, as the actio of U o Md {Z m is trasitive, it foows that r 0. Next, write ay A P U i the form A e it B, where t P R, B P SU. The for every z P C zt0u we have (2.5) Atrzsu pe it Bqtrzsu e it pbtrzsuq! e ittrcbc1 zsu e iσt d rt 2π CBC1 z. Represetig A as A e ipt`2πk `2πq pe2πik Bq with P Z, k P t0,...,1u, we obtai from (2.5) that σ must be of the form σ 1 ` pp ` q mq for some p,q P Z. This yieds (1.3). Simiary, the case whe SU acts as i (2.4) eads to (1.4). Proof of Propositio 1.5. We start by cosiderig actio (1.3) ad suppose first that it is ot effective. The for a otrivia eemet A e it B, t P R, B P SU oe has Atrzsu trzsu for a z P C zt0u. Hece, for some k P t0,..., 1u we have B e 2πik id, ad there exist P Z, K P t0,...,m 1u such that This yieds (2.6) e ip1`pp` q m qqt d rt 2πik 2π e t 2π r, k L r e 2πiK m d. 1 ` p ` q ` K m m

5 ACTIONS OF THE UNITARY GROUP ON QUOTIENTS OF HOPF MANIFOLDS 5 for some L P Z. Sice A is otrivia, we have r pp` q m q K m R Z, which is coditio (b). Coditio (a) foows from the secod idetity i (2.6). Coversey, suppose that for some P Z, K P t0,...,m 1u coditios (a) ad (b) hod. Defie t as i the first equatio i (2.6). Next, it foows from coditio (a) that the right-had side of the secod equatio i (2.6) is a iteger, thus we defie k by this equatio with L P Z chose arbitrariy. From coditio (b) we the immediatey see that A : e ipt`2πk q id is otrivia. O the other had, it is aso easy to check that A acts o Md triviay. Thus, the actio of U o Md {Z m is ot effective. The proof i the case of actio (1.4) is aaogous to the proof above. Refereces [BM] Bocher, S. ad Motgomery, D., Groups o aaytic maifods, A of Math. (2) 48 (1947), [IKra] Isaev, A. V. ad Kratz, S. G., O the automorphism groups of hyperboic maifods, J. Reie Agew. Math. 534 (2001), [IKru] Isaev, A. V. ad Kruzhii, N. G., Effective actios of the uitary group o compex maifods, Caadia J. Math. 54 (2002), Mathematica Scieces Istitute, Austraia Natioa Uiversity, Acto, ACT 2601, Austraia E-mai address: aexader.isaev@au.edu.au