Rank two nilpotent co-Higgs sheaves on complex surfaces

0.13 MB·English

by  Maurício Corrêa

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
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