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RANK TWO NILPOTENT CO-HIGGS SHEAVES ON COMPLEX SURFACES M.CORREˆA Dedicated to Jose Seade, forhis 60th birthday. 5 1 0 Abstract. Let (E,φ) be a rank two co-Higgs vector bundles on a K¨ahler 2 compactsurfaceX withφ∈H0(X,End(E)⊗TX)nilpotent. If(E,φ)issemi- v stable,thenoneofthefollowingholdsuptofinite´etalecover: o i) X isuniruled. N ii) X isatorusand(E,φ)isstrictlysemi-stable. iii) X isaproperlyellipticsurfaceand(E,φ)isstrictlysemi-stable. 8 1 ] G 1. Introduction. A AgeneralisedcomplexstructureonarealmanifoldX ofdimension2n,asdefined h. by Hitchin [10], is a rank-2n isotropic subbundle E0,1 ⊂(TX ⊕TX∗)C such that t i) E0,1⊕E0,1 =(T ⊕T∗)C a X X m ii) C∞(E0,1) is closed under the Courant bracket. On a manifold with a generalizedcomplex structure M. Gualtieri in [7] defined the [ notion of a generalized holomorphic bundle. More precisely, a generalized holo- 2 morphic bundle on a generalized complex manifold, is a vector bundle E with a v differential operator D : C∞(E) −→ C∞(E ⊗E0,1) such that for all smooth func- 5 4 tion f and all section s∈C∞(E) the following holds 0 i) D(fs)=∂(fs)+fD(s) 6 2 ii) D =0. 0 . In the case of an ordinary complex structure and D =∂+φ, for operators 1 0 ∂ :C∞(E)−→C∞(E ⊗T∗) 5 X 1 and : φ:C∞(E)−→C∞(E ⊗T ), v X i 2 2 X the vanishing D = 0 means that ∂ = 0 , ∂φ = 0 and φ∧φ = 0. By a classical 2 result of Malgrange the condition ∂ = 0 implies that E is a holomorphic vector r a bundle. On the other hand, ∂φ=0 implies that φ is a holomorphic global section φ∈H0(X,End(E)⊗T ) X which satisfies an integrability conditionφ∧φ=0. A co-Higgssheaf on a complex manifold X is a sheaf E together with a section φ ∈ H0(X,End(E)⊗T )(called X a Higgs fields) for which φ∧φ = 0. General properties of co-Higgs bundles were studied in [9, 20]. There is a motivation in physics for studying co-Higgs bundles, see [8], [12] and [26]. There are no stable co-Higgs bundles with nonzero Higgs field on curves C of genus g >1. (When g =1, a co-Higgs bundle is the same thing as a Higgs bundle Date:November 19,2015. 2010 Mathematics Subject Classification. Primary14D20-53D18, 14D06. Keywords and phrases. Higgsbundles-stablevector bundles. 1 2 M.CORREˆA in the usual sense.) In fact, contracting with a holomorphic differential gives a non-trivialendomorphismofE commuting with φ whichis impossible inthe stable case [9, 21] . S. Rayan showed in [19] the non-existence of stable co-Higgs bundles with non trivial Higgs field on K3 and general-type surfaces. In this note we prove the following result. Theorem 1.1. Let (E,φ) be a rank two co-Higgs vector bundles on a K¨ahler com- pact surface X with φ ∈ H0(X,End(E)⊗T ) nilpotent. If (E,φ) is semi-stable, X then one of the following holds up to finite ´etale cover: i) X is uniruled. ii) X is a torus and (E,φ) is strictly semi-stable. iii) X is a properly elliptic surface and (E,φ) is strictly semi-stable. ItfollowsdirectofproofofTheorem1.1thatthewecanconsideramoregeneral classes of singular projective surfaces . Theorem 1.2. Let (E,φ) be a rank two co-Higgs torsion-free sheaf on a normal projective surface X with φ ∈ H0(X,End(E)⊗T ) nilpotent. If (E,φ) is stable, X then X is uniruled. Finally,inthisworkweconsiderarelationbetweenco-HiggsbundlesandPoisson geometry on P1-bundles. In [23] Polishchuk associated to each rank-2 co-Higgs bundle (E,φ) a Poisson structure on its projectivized bundle P(E). This relation was explained by Rayan in [20] as follows: let Y := P(E) and consider the natural projection π : Y → X. The exact sequence 0−→T −→T −→π∗T −→0 X|Y Y X implies that T ⊗π∗T ⊂ 2T . Since T =Aut(P(E))=Aut(E)/C∗ we get X|Y X Y X|Y that V π (T ⊗π∗T )=π T ⊗T =End (E)⊗T , ∗ X|Y X ∗ X|Y X 0 X where End (E) denotes the trace-free endomorphisms of E. Therefore, we can as- 0 sociate a trace-free co-Higgs fields φ ∈ H0(X,End(E) ⊗ T ) a bi-vector π∗φ ∈ X H0(X,T ⊗π∗T )⊂H0(X, 2T ) on P(E). The co-Higgs condition φ∧φ=0 X|Y X Y implies that bi-vector π∗φ is intVegrable,see the introduction of [20]. The codimen- sion one foliation on P(E) is the called foliation by symplectic leaves induced by Poissonstruture . We get an interisting consequence of the proof Theorem 1.2. Corollary 1.1. If (E,φ) is locally free, stable and nilpotent , then the closure of the all leaves of the foliation by symplectic leaves on P(E) are rational surfaces. 2. Semi-stable co-Higgs sheaves Definition 2.1. A co-Higgs sheaf on a complex manifold X is a sheaf E together with a section φ∈H0(X,End(E)⊗T )(called a Higgs fields) for which φ∧φ=0. X Denote by End (E) := ker(tr : End(E) −→ O ) the trace-free part of the 0 X endomorphism bundle of E. Since End(E)=End (E)⊕O 0 X we have that End(E)⊗T = (End (E)⊗T )⊕T . Thus, the Higgs field φ ∈ X 0 X X H0(X,End(E)⊗T )canbe decomposedasφ=φ +φ ,where φ is the trace-free X 1 2 1 part and φ is a global vector field on X. In particular, if the surface X has no 2 global holomorphic vector fields, then every Higgs field is trace-free. RANK TWO NILPOTENT CO-HIGGS SHEAVES ON COMPLEX SURFACES 3 Definition 2.2. Let (X,ω) be a polarized K¨ahler compact manifold. We say that (E,φ) is semi-stable if c (F)·[ω] c (E)·[ω] 1 1 ≤ rank(F) rank(E) for all coherent subsheaves 06=F (E satisfying Φ(F)⊆F ⊗T , and stable if the X inequality is strict for all such F. We say that (E,φ) is strictly semi-stable if (E,φ) is semi-stable but non-stable. 3. Holomorphic foliations Definition 3.1. Let X be a connected complex manifold. A one-dimensional holo- morphic foliation is given by the following data: i) an open covering U ={U } of X; α ii) for each U an holomorphic vector field ζ ; α α iii) for every non-empty intersection, U ∩U 6=∅, a holomorphic function α β f ∈O∗ (U ∩U ); αβ X α β such that ζ =f ζ in U ∩U and f f =f in U ∩U ∩U . α αβ β α β αβ βγ αγ α β γ We denote by K the line bundle defined by the cocycle {f } ∈ H1(X,O∗). F αβ Thus,aone-dimensionalholomorphicfoliationF onX inducesaglobalholomorphic section ζ ∈ H0(X,T ⊗K ). The line bundle K is called the canonical bundle F X F F of F. Two sections ζ and η of H0(X,T ⊗K ) are equivalent, if there exists a F F X F never vanishing holomorphic function ϕ ∈H0(X,O∗), such that ζ = ϕ·η . It is F F clearthatζ andη define the samefoliation. Thus,a holomorphicfoliationF on F F X is an equivalence of sections of H0(X,T ⊗K ). X F 4. Examples 4.1. Canonical example of split co-Higgs bundles. Here we will give an ex- ample which naturally generalizes the canonical example given by Rayan on [20, Chapter 6]. Let (X,ω) be a polarized K¨ahler compact manifold. Suppose that thereexistsaglobalsectionζ ∈H0(X,Hom(N,TX⊗L))≃H0(X,TX⊗L⊗N∗). Now, consider the vector bundle E =L⊕N. Define the following co-Higgs fields φ:L⊕N −→(T ⊗L)⊕(T ⊗N)∈H0(X,End(E)⊗T ) X X X by φ(s,t) = (ζ(t),0). Since φ ◦ φ = 0 ∈ H0(End(E) ⊗ T ⊗ T ) we get that X X φ∧φ=0. Moreover,observe that the kernel of φ is the φ-invariant line bundle L. On the other hand, the line bundle L is destabilising only when [2c (L)−c (E)]·[ω]=[c (L)−c (N)]·[ω]>0. 1 1 1 1 That is, if [c (L)]·[ω]>[c (N)]·[ω]. 1 1 4.2. Co-Higgs bundles on ruled surfaces. Let C be a curve of genus g > 1. Now, consider the ruled surface X :=P(K ⊕O ) and C C π :P(K ⊕O )−→C C C the natural projection. Consider a Poisson structure on X given by a bivector σ ∈H0(X,∧2T ). Let (E,φ) be a nilpotent Higgs bundle on C. S. Rayan showed X in [20] that (π∗E,σ(π∗φ)) is a stable co-Higgs bundle on X. 4 M.CORREˆA 4.3. Co-Higgs orbibundles on weighted projective spaces. Let w ,w ,w 0 1 2 be positiveintegers,set|w|:=w +w +w . Assume that w ,w ,w arerelatively 0 1 2 0 1 2 prime. Define an action of C∗ in C3\{0} by C∗×(C3\{0}) −→ (C3\{0}) (4.1) λ.(z ,z ,z ) 7−→ (λw0z ,λw1z ,λw2z ) 0 1 2 0 1 2 and consider the weighted projective plane P(w ,w ,w ):=(C3\{0})/∼ 0 1 2 induced by the action above. We will denote this space by P(ω). On P(ω) we have an Euler sequence 2 ς 0−→OP(ω) −→ OP(ω)(ωi)−→TP(ω)−→0, Mi=0 whereOP(ω)isthetriviallineorbibundleandTP(ω)=Hom(Ω1P(ω),OP(ω))isthetan- gentorbibundleofP(ω). Themapς isgivenexplicitlybyς(1)=(ω z ,ω z ,ω z ). 0 0 1 1 2 2 Now,let (E,φ) be aco-Higgsorbibundle onP(ω). Tensoringthe Eulersequence by End(E), we obtain 2 0−→End(E)−→ End(E)(ω )−→End(E)⊗TP(ω)−→0. i Mi=0 Thus, the co-Higgs fields φ can be represented, in homogeneous coordinates, by ∂ ∂ ∂ φ=φ ⊗ +φ ⊗ +φ ⊗ , 0 1 2 ∂z ∂z ∂z 0 1 2 whereφ ∈H0(P(ω),End(E)(w )),foralli=0,1,2,andφ+θ⊗R definethesame i i ω co-Higgs field as φ, where R is the adapted radial vector field ω ∂ ∂ ∂ R =ω z +ω z ++ω z , ω 0 0 1 1 2 2 ∂z ∂z ∂z 0 1 2 with θ a endomorphism of E. Suppose that E =O(m )⊕O(m ) 1 2 and that there exists a stable φ for E such that m ≥m . Then 1 2 |m −m |≤ max {ω +ω }. 1 2 i j 0≤i6=j≤2 In fact, this is a consequence of Bott’s Formulae for weighted projective spaces. It follows from (see [6]) that H0(P(ω),TP(ω)⊗O (k))≃H0(P(ω),Ω1 ( 2 ω +k))6=∅ ω P(ω) i=0 i if and only if k > − max {ω +ω }. This generaliPze the example given by S. i j 0≤i6=j≤2 Rayan in [19]. 4.4. Co-Higgs bundles on two dimensional complex tori. Let X be a two dimensionalcomplex torus anda co-Higgsbundle φ∈H0(X,End(E)⊗T ). Then X φ is equivalente to a pair of commutative endomorphism of E. In fact, since the tangentbundle T isholomorphicallytrivial,wecantakeatrivializationbychoos- X ing two linearly independent globalvectorfields v ,v ∈H0(X,T ). Then, we can 1 2 X write φ=φ ⊗v +φ ⊗v . 1 1 2 2 The condition φ∧φ=0 implies that φ ◦φ =φ ◦φ . 1 2 2 1 RANK TWO NILPOTENT CO-HIGGS SHEAVES ON COMPLEX SURFACES 5 We have a canonicalnilpotent co-Higgs bundle (E,φ), where E =T =O⊕O and X 0 v , (cid:18) 0 0 (cid:19) where v is a global vector field on X. 5. Proof of Theorem Byusingthatthe Higgsfieldφ∈H0(X,End(E)⊗T )isnilpotentwehavethat X Ker(φ)=:L is a well defined line bundle on X . Thus , we have a exact sequence 0→L−→E −→I ⊗N −→0, Z where the nilpotent Higgs field φ factors as E −→ E ⊗T X ↓ ↑ (5.1) I ⊗N −→ L⊗T . Z X The morphism I ⊗N → L⊗T induces a holomorphic foliation on X which Z X induces a global section ζ ∈ H0(X,T ⊗L⊗N∗). Since det(E) = L⊗N we φ X conclude that L⊗N∗ =L2⊗det(E∗). Then ζ ∈H0(X,T ⊗L2⊗det(E∗)). φ X Let K := L2 ⊗det(E∗) the canonical bundle of the foliation F associated to the co-Higgs fields φ. If E is semi-stable then [c (K)]·[ω]=[2c (L)−c (E)]·[ω]≤0 1 1 1 for some K¨ahler class ω . If K ·[ω]<0, it follows from [14] that K is not pseudo- efective [5]. It follows from Brunella’s theorem [1] that X is uniruled. Now, suppose X is a normal projective surface and that K ·H = 0, for some H ample By Hodge index theorem we have that K2 ·H2 ≤ (K ·H)2 = 0, then K2 ≤ 0. Suppose that K2 < 0. We have that D = H +ǫK is a Q-divisor ample for 0<ǫ<<1, see [15, proposition 1.3.6]. Thus, we have that K·D =K·H +ǫ2K2 =ǫ2K2 <0. ByBogomolov-McQuillan-Miyaoka’stheorem[3]weconcludethatX isuniruled. If K2 =0, thenK is numericallytrivial. This factis wellknown,butfor convenience ofthereaderwegiveaproof. SupposethatthereexistsC ⊂X suchthatK·C >0. Now, Consider the divisor B = (H2)C −(H ·C)H. Then B·H = 0 and K ·B = (H2)K ·C < 0. Define F = mK +B for 0 < m << 1. Therefore F ·H = 0 and F2 > 0. This is a contradiction by the Hodge index Theorem. In this case E is strictly semi-stable. Now, we apply the classification, up to finite ´etale cover, of holomorphic folia- tions onprojective surfaces with canonicalbundle numerically trivial[22], [17], [2]. Therefore, up to finite ´etale cover, either: i) X is uniruled; ii) X is a torus; iii) k(X) = 1 and X = B ×C with g(B) ≥ 2, C is elliptic. That is, X is a sesquielliptic surface. IfX is K¨ahlerandnon-algebraicit followsfrom[2]that, upto finite´etalecover, either X has a unique elliptic fibration or X is a torus. 6 M.CORREˆA 6. Proof of Corollary 1.1 Since (E,φ) is nilpotent and stable the co-Higgs fields induces a foliation F by rationalcurvesonX. Now,considertheprojectivebundleπ :P(E)→X. Thenthe foliation by symplectic leaves G on P(E) is the pull-back of F by π. In particular, a closure of the a leaf of the foliation by symplectic leaves G is of type π−1(f(P1)), where f :P1 →X is the uniformization of a rationalleaf of F. Clearly π−1(f(P1)) is a rational surface. Acknowlegments. We are grateful to Arturo Fernandez-Perez, Renato Martins and Marcos Jardimforpointingoutcorrections. WearegratefultoHenriqueBursztynforinterestingconver- sationsaboutPoissonGeometry. References [1] M. Brunella, A positivity property for foliations on compact K¨ahler manifolds, Internat. J. Math.17(2006), 35-43. [2] M.Brunella,Foliationsoncomplex projectivesurfaces,inDynamicalSystems,PartII,Scuola Norm.Sup.,Pisa2003,49-77. [3] F.Bogomolov,M.McQuillan, Rationalcurvesonfoliatedvarieties,preprintIHESM/01/07 (2001) [4] K.Corlette, Flat G-bundles with canonical metrics.J.Differential Geom. 28,3(1988), 361- 382. [5] J.-P.Demailly,L2-vanishingtheoremsforpositivelinebundlesandadjunctiontheory,Tran- scendental Methods inAlgebraicGeometry,SpringerLectureNotes1646(1996), 1-97 [6] I.Dolgachev, Weighted projective varieties,Groupactions andvector fields,pp. 34-71,Lec- tureNotesinMathematics956,Springer-Verlag,Berlin-Heidelberg,NewYork(1982). [7] MGualtieri,Generalizedcomplexgeometry,Ann.ofMath.(2),2011arXiv:math/0401221v1. DPhilthesis,UniversityofOxford. [8] S. Gukov, E. Witten , Branes and quantization. Adv. Theor. Math. Phys. 13, 5 (2009), 1445-1518. [9] N.Hitchin, Generalized holomorphic bundles and the B-field action,J.Geom.andPhys.61 (2011), no.1,352-362. [10] N.Hitchin,Generalized Calabi-Yau manifolds. Q.J.Math.54,3(2003),281-308. [11] N. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55,1(1987), 59-126. [12] A. Kapustin , Y. Li, Open-string BRST cohomology for generalized complex branes. Adv. Theor.Math.Phys.9,4(2005), 559-574. [13] S. Kebekus, L.Sola-Conde, M. Toma. Rationally connected foliations after Bogomolov and McQuillan (mitLuisSolundMateiToma)JournalAlgebraicGeometry16 (2007), 65-81. [14] A.Lamari,Le cone K¨ahlerien d’une surface,J.Math.PuresAppl.78(1999), 249-263 [15] R.K.Lazarsfeld,Positivityinalgebraicgeometry,Vol.I,Springer,2000,ASeriesofModern SurveysinMathematics, No.48. [16] M. McQuillan, Noncommutative Mori theory, preprint IHES M/00/15 (2000) (re- vised: M/01/42(2001)) [17] M. McQuillan, Diophantine approximations and foliations, Publ. Math. IHES 87 (1998), 121-174 [18] Y. Miyaoka. Deformation of a morphism along a foliation. Algebraic Geometry Bowdoin 1985.Proc.Symp.PureMath.46(1987), 245-268. [19] Constructing co-Higgsbundles on CP2.Q.J.Math.65(2014), no.4,1437-1460 [20] S. Rayan, Geometry of co-Higgs bundles. D. Phil. thesis, Oxford, 2011. http://people.maths.ox.ac.uk/hitchin/hitchinstudents/rayan.pdf. [21] S.Rayan,Co-Higgsbundles on P1 ,NewYorkJ.Math.19(2013), 925-945; [22] T. Peternell, Generically nef vector bundles and geometric applications, in: Complex and differentialgeometry,345-368, SpringerProc.inMath.8,Springer-Verlag,Berlin(2011). [23] A.Polishchuk,Algebraic geometry of Poisson brackets.J.Math.Sci.84(1997), 1413-1444. [24] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I;Inst.HautesEtudes Sci.Publ.Math.,79(1994), 47-129. [25] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety II;Inst.Hautes EtudesSci.Publ.Math.,80(1994), 5-79. RANK TWO NILPOTENT CO-HIGGS SHEAVES ON COMPLEX SURFACES 7 [26] R.Zucchini,Generalizedcomplexgeometry,generalizedbranesandtheHitchinsigmamodel. J.HighEnergyPhys.,3(2005), 22-54 DepartamentodeMatema´tica-ICEX,UniversidadeFederaldeMinasGerais,UFMG Current address: Av. AntoˆnioCarlos6627,31270-901, BeloHorizonte-MG,Brasil. E-mail address: [email protected]

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