A Joint Adventure in Sasakian and Kähler Geometry

Transcription

1 A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015

2 2 Kähler Geometry Let N be a smooth compact manifold of real dimension 2d N. If J is a smooth bundle-morphism on the real tangent bundle, J : TN TN such that J 2 = Id and X, Y TN J(L X Y ) L X JY = J(L JX JY JL JX Y ), then (N, J) is a complex manifold with complex structure J. A Riemannian metric g on (N, J) is said to be a Hermitian Riemannian metric if X, Y TN, g(jx, JY ) = g(x, Y ) This implies that ω(x, Y ) := g(jx, Y ) is a J invariant (ω(jx, JY ) = ω(x, Y )) non-degenerate 2 form on N. If dω = 0, then we say that (N, J, g, ω) is a Kähler manifold (or Kähler structure) with Kähler form ω and Kähler metric g. The second cohomology class [ω] is called the Kähler class. For fixed J, the subset in H 2 (N, R) consisting of Kähler classes is called the Kähler cone.

3 3 Ricci Curvature of Kähler metrics: Given a Kähler structure (N, J, g, ω), the Riemannian metric g defines (via the unique Levi-Civita connection ) the Riemann curvature tensor R : TN TN TN TN and the trace thereoff, the Ricci tensor r : TN TN C (N) This gives us the Ricci form, ρ(x, Y ) = r(jx, Y ). The miracle of Kähler geometry is that c 1 (N, J) = [ ρ 2π ]. If ρ = λω, where λ is some constant, then we say that (N, J, g, ω) is Kähler-Einstein (or just KE). More generally, if ρ λω = L V ω, where V is a holomorphic vector field, then we say that (N, J, g, ω) is a Kähler-Ricci soliton (or just KRS). KRS = c 1 (N, J) is positive, negative, or null.

4 4 Scalar Curvature of Kähler metrics: Given a Kähler structure (N, J, g, ω), the Riemannian metric g defines (via the unique Levi-Civita connection ) the scalar curvature, Scal C (N), where Scal is the trace of the map X r(x ) where X, Y TN, g( r(x ), Y ) = r(x, Y ). If Scal is a constant function, we say that (N, J, g, ω) is a constant scalar curvature Kähler metric (or just CSC). KE = CSC (with λ = Scal 2d N ) Not all complex manifolds (N, J) admit CSC Kähler structures. There are generalizations of CSC, e.g. extremal Kähler metrics as defined by Calabi (L g ScalJ = 0). Not all complex manifolds (N, J) admit extremal Kähler structures either.

5 5 Admissible Kähler manifolds/orbifolds Special cases of the more general (admissible) constructions defined by/organized by Apostolov, Calderbank, Gauduchon, and T-F. Credit also goes to Calabi, Koiso, Sakane, Simanca, Pedersen, Poon, Hwang, Singer, Guan, LeBrun, and others. Let ω N be a primitive integral Kähler form of a CSC Kähler metric on (N, J). Let 1l N be the trivial complex line bundle. Let n Z \ {0}. Let L n N be a holomorphic line bundle with c 1 (L n ) = [n ω N ]. Consider the total space of a projective bundle S n = P(1l L n ) N. Note that the fiber is CP 1. S n is called admissible, or an admissible manifold.

6 6 Admissible Kähler classes Let D 1 = [1l 0] and D 2 = [0 L n ] denote the zero and infinity sections of S n N. Let r be a real number such that 0 < r < 1, and such that r n > 0. A Kähler class on S n, Ω, is admissible if (up to scale) Ω = 2πn[ω N ] r + 2πPD(D 1 + D 2 ). In general, the admissible cone is a sub-cone of the Kähler cone. In each admissible class we can now construct explicit Kähler metrics g (called admissible Kähler metrics). We can generalize this construction to the log pair (S n, ), where denotes the branch divisor = (1 1/m 1 )D 1 + (1 1/m 2 )D 2. If m = gcd(m 1, m 2 ), then (S n, ) is a fiber bundle over N with fiber CP 1 [m 1 /m, m 2 /m]/z m. g is smooth on S n \ (D 1 D 2 ) and has orbifold singularities along D 1 and D 2

7 7 Sasakian Geometry: Sasakian geometry: odd dimensional version of Kählerian geometry and special case of contact structure. A Sasakian structure on a smooth manifold M of dimension 2n + 1 is defined by a quadruple S = (ξ, η, Φ, g) where η is contact 1-form defining a subbundle (contact bundle) in TM by D = ker η. ξ is the Reeb vector field of η [η(ξ) = 1 and ξ dη = 0] Φ is an endomorphism field which annihilates ξ and satisfies J = Φ D is a complex structure on the contact bundle (dη(j, J ) = dη(, )) g := dη (Φ 1l) + η η is a Riemannian metric ξ is a Killing vector field of g which generates a one dimensional foliation F ξ of M whose transverse structure is Kähler. (Let (g T, ω T ) denote the transverse Kähler metric) (dt 2 + t 2 g, d(t 2 η)) is Kähler on M R + with complex structure I : IY = ΦY + η(y )t t for vector fields Y on M, and I (t t ) = ξ.

8 8 If ξ is regular, the transverse Kähler structure lives on a smooth manifold (quotient of regular foliation F ξ ). If ξ is quasi-regular, the transverse Kähler structure has orbifold singularities (quotient of quasi-regular foliation F ξ ). If not regular or quasi-regular we call it irregular... (that s most of them) Transverse Homothety: If S = (ξ, η, Φ, g) is a Sasakian structure, so is S a = (a 1 ξ, aη, Φ, g a ) for every a R + with g a = ag + (a 2 a)η η. So Sasakian structures come in rays.

9 9 Deforming the Sasaki structure: In its contact structure isotopy class: η η + d c φ, φ is basic This corresponds to a deformation of the transverse Kähler form ω T ω T + dd c φ in its Kähler class in the regular/quasi-regular case. Up to isotopy means that the Sasaki structure might have to been deformed as above.

10 10 In the Sasaki Cone: Choose a maximal torus T k, 0 k n + 1 in the Sasaki automorphism group Aut(S) = {φ Diff (M) φ η = η, φ J = J, φ ξ = ξ, φ g = g}. The unreduced Sasaki cone is t + = {ξ t k η(ξ ) > 0}, where t k denotes the Lie algebra of T k. Each element in t + determines a new Sasaki structure with the same underlying CR-structure.

11 11 Ricci Curvature of Sasaki metrics The Ricci tensor of g behaves as follows: r(x, ξ) = 2n η(x ) for any vector field X r(x, Y ) = rt (X, Y ) 2g(X, Y ), where X, Y are sections of D and r T is the transverse Ricci tensor If the transverse Kähler structure is Kähler-Einstein then we say that the Sasaki metric is η-einstein. S = (ξ, η, Φ, g) is η-einstein iff its entire ray is η-einstein ( η-einstein ray ) If the transverse Kähler-Einstein structure has positive scalar curvature, then exactly one of the Sasaki structures in the η-einstein ray is actually Einstein (Ricci curvature tensor a rescale of the metric tensor). That metric is called Sasaki-Einstein. If S = (ξ, η, Φ, g) is Sasaki-Einstein, then we must have that c 1 (D) is a torsion class (e.g. it vanishes).

12 12 A Sasaki Ricci Soliton (SRS) is a transverse Kähler Ricci soliton, that is, the equation ρ T λω T = L V ω T holds, where V is some transverse holomorphic vector field, and λ is some constant. So if V vanishes, we have an η-einstein Sasaki structure. Our definition allows SRS to come in rays. We will say that S = (ξ, η, Φ, g) is η-einstein / Einstein / SRS whenever it is η-einstein / Einstein /SRS up to isotopy.

13 13 Scalar Curvature of Sasaki metrics The scalar curvature of g behaves as follows Scal = Scal T 2n S = (ξ, η, Φ, g) has constant scalar curvature (CSC) if and only if the transverse Kähler structure has constant scalar curvature. S = (ξ, η, Φ, g) has CSC iff its entire ray has CSC ( CSC ray ). CSC can be generalized to Sasaki Extremal (Boyer, Galicki, Simanca) such that S = (ξ, η, Φ, g) is extremal if and only if the transverse Kähler structure is extremal S = (ξ, η, Φ, g) is extremal iff its entire ray is extremal ( extremal ray ). We will say that S = (ξ, η, Φ, g) is CSC/extremal whenever it is CSC/extremal up to isotopy.

14 14 The Join Construction The join construction of Sasaki manifolds (Boyer, Galicki, Ornea) is the analogue of Kähler products. Given quasi-regular Sasakian manifolds π i : M i Z i. Let L = 1 2l 1 ξ 1 1 2l 2 ξ 2. Form (l 1, l 2 )- join by taking the quotient by the action induced by L: M 1 M 2 π L π12 M 1 l1,l 2 M 2 Z 1 Z 2 π M 1 l1,l 2 M 2 is a S 1 -orbibundle (generalized Boothby-Wang fibration). M 1 l1,l 2 M 2 has a natural quasi-regular Sasakian structure for all relatively prime positive integers l 1, l 2. Fixing l 1, l 2 fixes the contact orbifold. It is a smooth manifold iff gcd(µ 1 l 2, µ 2 l 1 ) = 1, where µ i is the order of the orbifold Z i.

15 15 Join with a weighted 3-sphere Take π 2 : M 2 Z 2 to be the S 1 -orbibundle π 2 : S 3 w CP[w] determined by a weighted S 1 -action on S 3 with weights w = (w 1, w 2 ) such that w 1 w 2 are relative prime. S 3 w has an extremal Sasakian structure. Let M 1 = M be a regular CSC Sasaki manifold whose quotient is a compact CSC Kähler manifold N. Assume gcd(l 2, l 1 w 1 w 2 ) = 1 (equivalent with gcd(l 2, w i ) = 1). M Sw 3 π L π 12 M l1,l 2 Sw 3 =: M l1,l 2,w π N CP[w]

16 16 The w-sasaki cone The Lie algebra aut(s l1,l 2,w) of the automorphism group of the join satisfies aut(s l1,l 2,w) = aut(s 1 ) aut(s w ), mod (L l1,l 2,w = 1 2l 1 ξ 1 1 2l 2 ξ 2 ), where S 1 is the Sasakian structure on M, and S w is the Sasakian structure on Sw. 3 The unreduced Sasaki cone t + l 1,l 2,w of the join M l 1,l 2,w thus has a 2-dimensional subcone t + w is called the w-sasaki cone. t + w is inherited from the Sasaki cone on S 3 Each ray in t + w is determined by a choice of (v 1, v 2 ) R + R +. The ray is quasi-regular iff v 2 /v 1 Q. t + w has a regular ray (given by (v 1, v 2 ) = (1, 1)) iff l 2 divides w 1 w 2.

17 17 Motivating Questions Does t + w have a CSC/η-Einstein ray? What about extremal/sasaki-ricci solitons?

18 18 Key Proposition (Boyer, T-F) Let M l1,l 2,w = M l1,l 2 Sw 3 be the join as described above. Let v = (v 1, v 2 ) be a weight vector with relatively prime integer components and let ξ v be the corresponding Reeb vector field in the Sasaki cone t + w. Then the quotient of M l1,l 2,w by the flow of the Reeb vector field ξ v is (S n, ) ( with n = l w1v ) 2 w 2v 1 1 s, where s = gcd(l2, w 1 v 2 w 2 v 1 ), and is the branch divisor = (1 1 )D 1 + (1 1 )D 2, (1) m 1 m 2 with ramification indices m i = v i l 2s.

19 19 The Kähler class on the (quasi-regular) quotient is admissible up to scale. We can determine exactly which one it is. So we can test it for containing admissible KRS, KE, CSC, or extremal metrics. Hence we can test if the ray of ξ v is (admissible and) η-einstein/srs/csc/extremal (up to isotopy). By lifting the admissible construction to the Sasakian level (in a way so it depends smoothly on (v 1, v 2 )), we can also handle the irregular rays.

20 20 Theorem A (Boyer, T-F) For each vector w = (w 1, w 2 ) Z + Z + with relatively prime components satisfying w 1 > w 2 there exists a Reeb vector field ξ v in the 2-dimensional w-sasaki cone on M l1,l 2,w such that the corresponding ray of Sasakian structures S a = (a 1 ξ v, aη v, Φ, g a ) has constant scalar curvature. Suppose in addition that the scalar curvature of N is non-negative. Then the w-sasaki cone is exhausted by extremal Sasaki metrics. In particular, if the Kähler structure on N admits no Hamiltonian vector fields, then the entire Sasaki cone of the join M l1,l 2,w can be represented by extremal Sasaki metrics. Suppose in addition that the scalar curvature of N is positive. Then for sufficiently large l 2 there are at least three CSC rays in the w-sasaki cone of the join M l1,l 2,w.

21 21 Theorem B (Boyer, T-F) Suppose N is positive Kähler-Einstein with Fano index I N and l 1 = I N gcd(w 1 + w 2, I N ), l w 1 + w 2 2 = gcd(w 1 + w 2, I N ), (ensures that c 1 (D) vanishes). Then for each vector w = (w 1, w 2 ) Z + Z + with relatively prime components satisfying w 1 > w 2 there exists a Reeb vector field ξ v in the 2-dimensional w-sasaki cone on M l1,l 2,w such that the corresponding Sasakian structure S = (ξ v, η v, Φ, g) is Sasaki-Einstein. Moreover, this ray is the only admissible CSC ray in the w-sasaki cone. In addition, for each vector w = (w 1, w 2 ) Z + Z + with relatively prime components satisfying w 1 > w 2 every single ray in the 2-dimensional w-sasaki cone on M l1,l 2,w admits (up to isotopy) a Sasaki-Ricci soliton.

22 22 Remarks The Sasaki-Einstein structures were first found by the physicists Guantlett, Martelli, Sparks, Waldram. Starting from the join construction allows us to study the topology of the Sasaki manifolds more closely. When N = CP 1, M l1,l 2,w are S 3 -bundles over S 2. These were treated by Boyer and Boyer, Pati, as well as by E. Legendre. Our set-up, starting from a join construction, allows for cases where no regular ray in the w-sasaki cone exists. If, however, the given w-sasaki cone does admit a regular ray, then the transverse Kähler structure is a smooth Kähler Ricci soliton and the existence of an SE metric in some ray of the Sasaki cone is predicted by the work of Mabuchi and Nakagawa.

23 23 References Apostolov, Calderbank, Gauduchon, and T-F. Hamiltonian 2-forms in Kähler geometry, III Extremal metrics and stability, Inventiones Mathematicae 173 (2008), For the admissible construction of Kähler metrics Boyer and Galicki Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, Other papers by Boyer et all. For the join of Sasaki structures Boyer and T.-F. The Sasaki Join, Hamiltonian 2-forms, and Constant Scalar Curvature (to appear in JGA, 2015) and references therein to our previous papers. For the details and proofs behind the statements in this talk.