INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS

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1 INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable We also show that such a manfold s weakly (C ) n -equvarantly somorphc to a torc manfold 1 Introducton A torc manfold s a nonsngular complete torc varety As a topologcal analogue of a torc manfold, the noton of topologcal torc manfold has been ntroduced by the author, Y Fukukawa and M Masuda [3] A topologcal torc manfold of dmenson 2n s a smooth closed manfold endowed wth an effectve (C ) n -acton havng an open dense orbt, and locally equvarantly dffeomorphc to a smooth representaton space of (C ) n We note that a topologcal torc manfold s locally equvarantly dffeomorphc to an algebrac representaton space f and only f t s a torc manfold A quastorc manfold ntroduced by M Davs and T Januskewcz [1] of dmenson 2n s a smooth closed manfold endowed wth a locally standard (S 1 ) n -acton, whose orbt space s a smple polytope In [3], t s shown that any quastorc manfold s a topologcal torc manfold wth the restrcted compact torus acton Conversely, t s also shown that any topologcal torc manfold of dmenson less than or equal to 6 wth the restrcted compact torus acton s a quastorc manfold However, there are nfntely many topologcal torc manfolds wth the restrcted compact torus acton whch are not equvarantly dffeomorphc to any quastorc manfold Among quastorc manfolds, some admt nvarant almost complex structures under the compact torus actons M Masuda provded examples of 4-dmensonal quastorc manfolds whch admt (S 1 ) 2 -nvarant almost complex structures (see [4, Theorem 51]) A Kustarev descrbed a necessary and suffcent condton for a quastorc manfold to admt a torus nvarant almost complex structure for arbtrary dmenson (see [2, Theorem 1]) As we mentoned, any quastorc manfold s a topologcal torc manfold wth the restrcted compact torus acton In ths paper, we dscuss on (C ) n -nvarant stably, or almost complex structures on topologcal torc manfolds of dmenson 2n The followngs are our results: Theorem 11 Let X be a topologcal torc manfold of dmenson 2n Let R 2l be the product bundle of rank 2l over X, T X the tangent bundle of X If there Date: February 21, Mathematcs Subect Classfcaton 32Q60, 57S20, 57S25 Key words and phrases torc manfold, quastorc manfold 1

2 2 HIROAKI ISHIDA exsts a (C ) n -nvarant stably complex structure J on T X R 2l, then T X becomes a complex subbundle of T X R 2l Namely, X has an nvarant almost complex structure Theorem 12 Let X be a topologcal torc manfold of dmenson 2n, J a (C ) n - nvarant almost complex structure Then, J s ntegrable and X s weakly equvarantly somorphc to a torc manfold Namely, there are a torc manfold Y, a bholomorphsm f : X Y and a smooth automorphsm ρ of (C ) n such that f g = ρ(g) f for all g (C ) n If we replace the condton (C ) n -nvarant by (S 1 ) n -nvarant on the almost complex structure J, then Theorem 12 does not hold For example, CP 2 #CP 2 #CP 2 wth an effectve (S 1 ) 2 -acton s a topologcal torc manfold wth the restrcted (S 1 ) 2 -acton One can show that there exsts an (S 1 ) 2 -nvarant almost complex structure on CP 2 #CP 2 #CP 2 (see [4, Theorem 51]) However, CP 2 #CP 2 #CP 2 carres no complex structure because CP 2 #CP 2 #CP 2 does not ft Kodara s classfcaton of complex surfaces Namely, the almost complex structure s not ntegrable For a topologcal torc manfold X of dmenson 2n, there s a canoncal short exact sequence of (C ) n -bundles m 0 C m n L T X 0, =1 where L s are complex lne bundles (see [3, Theorem 61]) Theorems 11 and 12 say that the short exact sequence above does not splt as (C ) n -bundles unless X s a torc manfold 2 Prelmnares In ths secton, we revew the quotent constructon of topologcal torc manfolds and the correspondence between topologcal torc manfolds and nonsngular complete topologcal fans (see [3] for detals) A nonsngular complete topologcal fan s a par = (Σ, β) such that (1) Σ s an abstract smplcal complex on [m] = {1,, m}, (2) β : [m] (C Z) n s a functon whch satsfes the followng: (a) Let Re be the composton of two natural proectons (C Z) n C n and C n R n We assgn a cone { } a (Re β)() a 0 to each smplex I n Σ Then, we have a collecton of cones n R n () Each par of two cones does not overlap on ther relatve nterors Namely, the real part Re β of β together wth Σ forms an ordnary fan () The unon of all cones concdes wth R n Namely, the fan s complete (b) The nteger part of β together wth Σ forms a nonsngular mult-fan (see [4, p249])

3 INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS 3 It follows from (2a) that Σ must be a smplcal (n 1)-sphere wth m vertces If we regard ntegers Z as a subset of C Z va a (a, a) for a Z, then any nonsngular complete fan can be regarded as a specal case of a nonsngular complete topologcal fan Conversely, f the mage of β s contaned n the dagonal subgroup Z n of (C Z) n, then becomes a nonsngular complete fan We express β() as β = (β 1,, βn ) (C Z)n and β = (b + 1c, v ) C Z for = 1,, m and = 1,, n For a nonsngular complete topologcal fan = (Σ, β), we can construct a topologcal torc manfold as follows We set for I [m], and U(I) := {(z 1,, z m ) C m z 0 for / I} U(Σ) := U(I) I Σ We defne a group homomorphsm λ β : C (C ) n by (21) λ β (h ) := (h β1,, h βn ), where (22) h β := h b + 1c ( ) v h h We defne a group homomorphsm λ : (C ) m (C ) n by m λ(h 1,, h m ) := λ β (h ) One can show that λ s surectve (see [3, Lemma 41]) We note that the (C ) m - acton on U(Σ) gven by coordnatewse multplcatons nduces the acton of (C ) m / ker λ on the quotent space X( ) := U(Σ)/ ker λ Snce λ s surectve, we can dentfy (C ) m / ker λ wth (C ) n through λ Hence X( ) s equpped wth the (C ) n -acton One can show that X( ) s a topologcal torc manfold We shall remember the equvarant charts and transton functons of X( ) descrbed n [3] for later use We regard β = (b + 1c, v ) as the followng matrx: β = ( b =1 0 c v And we also regard β as an n-tuple (β 1,, βn ) of square matrces of sze 2 Let Σ (n) denote the set of (n 1)-dmensonal smplces n Σ For I Σ (n), the dual {α I} of {β } s defned to be (23) α I h, β = δ h1, ) where δ denotes the the Kronecker delta, and, s gven by n α, β = α β for n-tuples α = (α 1,, α n ) and β = (β 1,, β n ) of square matrces of sze 2 The equvarant charts are gven as follows For α = (α 1,, α n ), we defne a =1

4 4 HIROAKI ISHIDA representaton χ α : (C ) n C by χ α (g 1,, g n ) := n =1 g α Let V (χ α ) denote the representaton space of χ α For I Σ (n), the equvarant chart ϕ I : U(I)/ ker λ V (χαi ) s defned by m ϕ I ([z 1,, z m ]) :=, =1 z αi,β where [z 1,, z m ] denotes the equvalence class of (z 1,, z m ) U(Σ) Let (w ) be the coordnates of V (χαi ) Namely, w := m =1 z αi,β An omnorentaton of a topologcal torc manfold X s a choce of orentatons of normal bundles of characterstc submanfolds of X Here, a characterstc submanfold of X s a connected (C ) n -nvarant submanfold of codmenson 2 By the constructon above, β allows us to decde an omnorentaton of X( ) as follows Each characterstc submanfold ntersectng U(I)/ ker λ s locally represented as {(w ) w 0 = 0} for some 0 I through ϕ I : U(I)/ ker λ V (χαi ) We choose the orentaton of the normal bundle of each characterstc submanfold so that ϕ I preserves the orentatons of the normal bundles of the characterstc submanfolds and the normal bundle of {(w ) w 0 = 0} n V (χαi ) We need to show that ths s well-defned By (23), λ β0 (C ) fxes {(w ) w 0 = 0} Let us consder the dfferental (λ β0 ( 1)) of the acton λ β0 ( 1) at a pont of {(w ) w 0 = 0} The normal vector space of {(w ) w 0 = 0} n V (χαi ) at any pont n {(w ) w 0 = 0} can be dentfed wth V (χ αi 0 ) By drect computaton, we have χ αi 0 λ β0 ( 1) = 1 (see [3, Lemma 22(2)]) Namely, for any pont of the characterstc submanfold and any nonzero normal vector ξ at the pont, f we choose the orentaton of the normal bundle of the characterstc submanfold so that (ξ, (λ β0 ( 1)) (ξ)) s a postve bass, then ϕ I preserves the orentatons of the normal bundles The correspondence X( ) s bectve between nonsngular complete topologcal fans and omnorented topologcal torc manfolds (see [3, Theorem 81]) We shall see the nverse correspondence For a topologcal torc manfold X of dmenson 2n wth an omnorentaton, let us denote characterstc submanfolds of X by X 1,, X m Defne { } Σ = I [m] X For an orentaton on normal bundle of X, we can fnd a unque complex structure J such that the orentaton concdes wth the orentaton whch comes from J,

5 INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS 5 the C -subgroup of (C ) n whch fxes each pont of X acts on the normal bundle as C-lnear wth respect to J For J, we can fnd a unque β (C Z) n such that (λ β (h)) (ξ) = hξ for any normal vector ξ and h C, where the rght hand sde s the multplcaton wth complex number For an omnorentaton of X, defne β : [m] (C Z) n as β() := β Then, the par (X) = (Σ, β) becomes a nonsngular complete topologcal fan and the correspondence X (X) s the nverse correspondence of X( ) Namely, there exsts an equvarant dffeomorphsm X X( (X)) whch preserves the omnorentatons The transton functons of X( ) are gven as follows Let K be another element n Σ (n) By drect computaton, k-component of ϕ K (ϕ 1 I (w ) ) for k K s gven as (24) (see [3, Lemma 52]) We remark that ( w w αk k,β w αk k,β f and only f αk K, β = γk, K 1 for some nteger γk k, (see (22)) Ths mples that all transton functons are holomorphc f and only f there s an nteger γk, K such that αk K, β = γk, K 1 for all [m], k K and K Σ(n) In ths case, each transton functon s a Laurent monomal and hence X( ) s weakly equvarantly dffeomorphc to a torc manfold ) = 0 3 Proof of Theorem 11 Let X be a 2n-dmensonal topologcal torc manfold, T X the tangent bundle of X, J a (C ) n -nvarant complex structure on T X R 2l We take an omnorentaton of X and consder the topologcal fan = (Σ, β) assocated to X = X( ) wth the gven omnorentaton We defne a cross secton e h : X R 2l = X R 2l for h = 1,, 2l by x (x, e h ) for all x X, where e h denotes the h-th standard bass vector of R 2l There s a natural ncluson (C ) n X gven by g g [1,, 1] where [1,, 1] denotes the equvalence class of (1,, 1) n U(Σ) For I Σ (n), the ncluson s of the form ) (31) χ αi : g = (g ) =1,,n (χ αi (g) va the equvarant local chart ϕ I : U I / ker λ V (χαi ) We dentfy V (χαi ) wth R 2n by (32) w = x + 1y for I, where (w ) denote the coordnates of V (χαi ) We also dentfy (C ) n wth (R R/2πZ) n by ( (33) ψ : (g ) =1,,n log g, ( )) g 1 log g =1,,n

6 6 HIROAKI ISHIDA Let (τ, θ ) =1,,n be the coordnates of (R R/2πZ) n Snce J s (C ) n -nvarant, the matrx representaton, denoted J 0, of J on (R R/2πZ) n wth respect to the coordnates (τ, θ ) =1,,n and sectons e h s s constant Let Ψ I : (R 2 \ {0}) n (R R/2πZ) n be the composton of the dentfcaton (32), the nverse of (31), and ψ Namely, ( ) 1 (34) Ψ I ((x, y ) ) := ψ χ αi ((x + 1y ) ) Snce ( χαi ) 1 concdes wth λ β (see [3, Lemma 23]), t follows from (21) and (22) that the coordnates (τ, θ ) =1,,n are represented as ( τ = log (x + ) 1y ) β and = 1 b 2 log(x2 + y 2 ) θ = ( (x + ) 1y ) β 1 log (x + 1y ) β = 1 log x + ( 1y 1c x + ) v 1y x + 1y = ( c + 1v log(x 2 + y 2 ) 1v 2 log(x + ) 1y ) Then, by drect computaton, we have τ x = b x x 2 +, y2 τ y = b y x 2 +, y2 θ = (c + 1v )x x x 2 + 1v 1 y2 x + 1y and Therefore, where ( τ x θ x = c x v y x 2 + y2 θ = (c + 1v )y y x v 1 y2 x + 1y τ y θ y = c y + v x x 2 + y2 ) ( b = 0 c v ) ( x x 2 +y2 y x 2 +y2 y x 2 +y2 x x 2 +y2 ) = β t, ( ) 1 x y t = x 2 + C GL(2, R) y2 y x

7 INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS 7 We set two square matrces B = (β ), and T = dag(t ; I) of sze n whose entres are square matrces of sze 2 Then, the dfferental T (x,y) Ψ I of Ψ I at (x, y) s represented as BT wth respect to the coordnates (x, y ) and (τ, θ ) =1,,n Hence the complex structure J of T X R 2l s represented on V (χαi ) = R 2n as the followng square matrx ( ) ( ) (BT ) 1 BT (35) J I := J I 0 2l I 2l of sze 2n + 2l wth respect to the coordnates (x, y ) and sectons e h, where I 2l denote the dentty matrx of sze 2l We set ( ) J11 J J 0 =: 12 J 21 J 22 where J 11, J 12, J 21, J 22 are matrces of 2n 2n, 2n 2l, 2l 2n, 2l 2l, respectvely Then, ( ) T (36) J I = 1 (B 1 J 11 B)T T 1 B 1 J 12 J 21 BT J 22 Each entry of J I s a homogeneous (ratonal) polynomal n (x, y ) However, each entry of J I must be a smooth functon on R 2n, n partcular, at the orgn Hence each entry of J 21 must be 0 Ths means that the tangent space at any pont of X s stable under J It follows that T X s a complex subbundle of T X R 2l wth respect to J The theorem s proved 4 Proof of Theorem 12 Let X be a topologcal torc manfold of dmenson 2n wth a (C ) n -nvarant almost complex structure J Snce any characterstc submanfold of X and X tself are almost complex submanfolds, the normal bundles of characterstc submanfolds of X become complex lne bundles Hence, we have a topologcal fan = (Σ, β) assocated to X Namely, for each characterstc submanfold X of X, we choose the unque β (C Z) n so that λ β (C ) fxes all ponts n X, and (λ β (h)) (ξ) = hξ for any h C and any normal vector ξ of X Accordng to the proof of Theorem 11, we dentfy the dense orbt wth (R R/2πZ) n and V (χαi ) wth R 2n for I Σ (n) Snce J s (C ) n -nvarant, the matrx representaton, denoted J 0, of J on (R R/2πZ) n wth respect to the coordnate (τ, θ ) =1,,n of (R R/2πZ) n s constant Let us remnd Ψ I : (R 2 \ {0}) n (R R/2πZ) n (see (34)) Then, the almost complex structure J s represented on V (αi ) = R2n as the followng square matrx (41) J I := (BT ) 1 J 0 (BT ) = T 1 (B 1 J 0 B)T (ths s the case when l = 0 n (35) and (36)) As we saw at the end of Secton 3, each entry of J I s a homogeneous (ratonal) polynomal n (x, y ) However, each entry of J I must be a smooth functon on R 2n, n partcular, at the orgn Hence J I must be constant Snce J I and B 1 J 0 B are both constant and T can take any element n (C ) n, t follows from (41) that

8 8 HIROAKI ISHIDA J I and B 1 J 0 B must be both n (C ) n and hence J I = B 1 J 0 B Moreover, JI 2 s the mnus dentty matrx because J I s the matrx representng the almost complex structure J It follows that J I must be of the form ( ) 0 1 ± 1 0 ( ) 0 1 ± 1 0 However, each β s taken so that (λ β (h)) (ξ) = hξ for any h C and any normal vector ξ of X, as we mentoned n Secton 2, we have ( ) J I = ( ) Clearly, the complex structure J I on R 2n comes from the dentfcaton (32) Therefore, J s ntegrable and the local chart ϕ I : U I / ker λ V (χαi ) s a holomorphc chart Ths mples that for another smplex K Σ (n), k-component of ϕ K (ϕ 1 I (w ) ) gven as (24) for k K s holomorphc Thus, the transton functons must be Laurent monomals as remarked at the end of Secton 2 and hence X( ) s weakly equvarantly somorphc to a torc manfold The theorem s proved Acknowledgement The author would lke to thank Professor Mkya Masuda for stmulatng dscusson and helpful comments on the presentaton of the paper References [1] M Davs and T Januskewcz, Convex polytopes, Coxeter orbfolds and torus actons, Duke Math J 62 (1991), [2] A Kustarev, Equvarant almost complex structures on quastorc manfolds, Russan Math Surveys 64 (2009), no 1, [3] H Ishda, Y Fukukawa and M Masuda, Topologcal torc manfolds, preprnt, arxv:mathat/ [4] M Masuda, Untary torc manfolds, mult-fans and equvarant ndex, Tohoku Math J 51 (1999), Department of Mathematcs, Graduate School of Scence, Osaka Cty Unversty, Sugmoto, Sumyosh-ku, Osaka , Japan E-mal address: hroakshda86@gmalcom