A Joint Adventure in Sasakian and K ahler Geometry Charles Boyer - PowerPoint PPT Presentation

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

2 K¨ ahler Geometry Let N be a smooth compact manifold of real dimension 2 d 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 − J L 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 d ω = 0, then we say that ( N , J , g , ω ) is a K¨ ahler manifold (or K¨ ahler structure ) with K¨ ahler form ω and K¨ ahler metric g . ◮ The second cohomology class [ ω ] is called the K¨ ahler class . ◮ For fixed J , the subset in H 2 ( N , R ) consisting of K¨ ahler classes is called the K¨ ahler cone .

3 Ricci Curvature of K¨ ahler metrics: Given a K¨ ahler 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 K¨ ahler geometry is that c 1 ( N , J ) = [ ρ 2 π ]. ◮ If ρ = λω, where λ is some constant, then we say that ( N , J , g , ω ) is K¨ ahler-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 K¨ ahler-Ricci soliton (or just KRS ). ◮ KRS = ⇒ c 1 ( N , J ) is positive, negative, or null.

4 Scalar Curvature of K¨ ahler metrics: Given a K¨ ahler 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 K¨ ahler metric (or just CSC ). ◮ KE = ⇒ CSC (with λ = Scal 2 d N ) ◮ Not all complex manifolds ( N , J ) admit CSC K¨ ahler structures. ◮ There are generalizations of CSC, e.g. extremal K¨ ahler metrics as defined by Calabi ( L ∇ g Scal J = 0) . ◮ Not all complex manifolds ( N , J ) admit extremal K¨ ahler structures either.

5 Admissible K¨ ahler 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 K¨ ahler form of a CSC K¨ ahler metric on ( N , J ). ◮ Let 1 l → 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 ( 1 l ⊕ L n ) → N . ◮ Note that the fiber is CP 1 . ◮ S n is called admissible , or an admissible manifold .

6 Admissible K¨ ahler classes ◮ Let D 1 = [ 1 l ⊕ 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 K¨ ahler class on S n , Ω, is admissible if (up to scale) Ω = 2 π n [ ω N ] + 2 π PD ( D 1 + D 2 ). r ◮ In general, the admissible cone is a sub-cone of the K¨ ahler cone. ◮ In each admissible class we can now construct explicit K¨ ahler metrics g (called admissible K¨ ahler 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 Sasakian Geometry: Sasakian geometry: odd dimensional version of K¨ ahlerian geometry and special case of contact structure . A Sasakian structure on a smooth manifold M of dimension 2 n + 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 ⌋ d η = 0] ◮ Φ is an endomorphism field which annihilates ξ and satisfies J = Φ | D is a complex structure on the contact bundle ( d η ( J · , J · ) = d η ( · , · )) ◮ g := d η ◦ (Φ ⊗ 1 l ) + 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 K¨ ahler. ◮ (Let ( g T , ω T ) denote the transverse K¨ ahler metric) ◮ ( dt 2 + t 2 g , d ( t 2 η )) is K¨ ahler on M × R + with complex structure I : IY = Φ Y + η ( Y ) t ∂ ∂ t for vector fields Y on M , and I ( t ∂ ∂ t ) = − ξ .

8 ◮ If ξ is regular , the transverse K¨ ahler structure lives on a smooth manifold (quotient of regular foliation F ξ ). ◮ If ξ is quasi-regular , the transverse K¨ ahler 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, a η, Φ , g a ) for every a ∈ R + with g a = ag + ( a 2 − a ) η ⊗ η . ◮ So Sasakian structures come in rays.

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 K¨ ahler form ω T → ω T + dd c φ in its K¨ ahler class in the regular/quasi-regular case. ◮ “Up to isotopy” means that the Sasaki structure might have to been deformed as above.

10 In the Sasaki Cone: ◮ Choose a maximal torus T k , 0 ≤ k ≤ n + 1 in the Sasaki automorphism group Aut ( S ) = { φ ∈ D iff ( 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 Ricci Curvature of Sasaki metrics ◮ The Ricci tensor of g behaves as follows: ◮ r ( X , ξ ) = 2 n η ( X ) for any vector field X ◮ r ( X , Y ) = r T ( X , Y ) − 2 g ( X , Y ), where X , Y are sections of D and r T is the transverse Ricci tensor ◮ If the transverse K¨ ahler structure is K¨ ahler-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 K¨ ahler-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 ◮ A Sasaki Ricci Soliton (SRS) is a transverse K¨ ahler 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 Scalar Curvature of Sasaki metrics ◮ The scalar curvature of g behaves as follows Scal = Scal T − 2 n ◮ S = ( ξ, η, Φ , g ) has constant scalar curvature (CSC) if and only if the transverse K¨ ahler 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 K¨ ahler 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 The Join Construction ◮ The join construction of Sasaki manifolds (Boyer, Galicki, Ornea) is the analogue of K¨ ahler products. ◮ Given quasi-regular Sasakian manifolds π i : M i → Z i . Let 1 1 L = 2 l 1 ξ 1 − 2 l 2 ξ 2 . ◮ Form ( l 1 , l 2 )- join by taking the quotient by the action induced by L : M 1 × M 2 ց π L   M 1 ⋆ l 1 , l 2 M 2  π 12  � ւ π Z 1 × Z 2 ◮ M 1 ⋆ l 1 , l 2 M 2 is a S 1 -orbibundle (generalized Boothby-Wang fibration). ◮ M 1 ⋆ l 1 , 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 .