Exotic nearly Kähler structures on S 6 and S 3 S 3

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1 Exotic nearly Kähler structures on S 6 and S 3 S 3 Lorenzo Foscolo Stony Brook University joint with Mark Haskins, Imperial College London Friday Lunch Seminar, MSRI, April

2 G 2 cones and nearly Kähler 6 manifolds (M 6, g) smooth Riemannian manifold Riemannian cone over M: C(M) = R + M endowed with the (incomplete) Riemannian metric dr 2 + r 2 g Hol(C) G 2 parallel forms ϕ = r 2 dr ω + r 3 ReΩ, ϕ = r 3 dr ImΩ r 4 ω 2 dϕ = 0 = d ϕ the SU(3) structure (ω, Ω) on M satisfies { dω = 3ReΩ dimω = 2ω 2 (NK) A 6 manifold M endowed with an SU(3) structure satisfying (NK) is called a nearly Kähler (nk) 6 manifold. nk manifolds and real Killing spinors nk manifolds and Gray Hervella classes of almost Hermitian manifolds every nk 6 manifold M is Einstein with Scal = 30 = M is compact with π 1 (M) < = we will assume M simply connected

3 The four known examples 1951, Steenrod s book: S 6 ImO 1968, Gray Wolf s classification of 3 symmetric spaces: S 3 S 3 = SU(2) 3 / SU(2) CP 3 = Sp(2)/U(1) Sp(1) F 3 = SU(3)/T 2 3 symmetric space: G/K, where G is a compact semisimple Lie group and K is the fixed point set of an automorphism of G of order 3 CP 3 and F 3 are the twistor spaces of S 4 and CP 2 respectively Connection with G 2 holonomy noted only in the 1980 s, e.g. Bryant s 1987 first example of a full G 2 holonomy metric is C(F 3 ) G 2 cones as local models for isolated singularities of G 2 manifolds Infinitely many Calabi Yau, hyperkähler and Spin(7) cones 2005, Butruille: the four known examples are the only homogeneous nk 6 manifolds 2006, Bryant: local generality of nk structures on 6 manifolds the same as for Calabi Yau structures

4 Main Theorem and possible proof strategies Main Theorem (F. Haskins, 2015) There exists an inhomogeneous nearly Kähler structure on S 6 and S 3 S 3. Two natural strategies to find nk 6 manifolds: Symmetries: cohomogeneity one nk 6 manifolds Desingularisation of nk sine-cones Both play a role in the proof of the Main Theorem Simplest singular nk spaces: cross-sections of split G 2 cones R C for a Calabi Yau cone C. (N 5, g N ) smooth Sasaki Einstein, i.e. C(N) is a Calabi Yau (CY) cone The sine-cone over N: SC(N) = [0, π] N endowed with the Riemannian metric dr 2 + sin 2 r g N SC(N) is nk but has two isolated singularities each modelled on C(N) Try to desingularise SC(N) by gluing complete asymptotically conical (AC) CY 3 folds

5 The simplest nk sine-cone and desingularisations N = N 1,1 := SU(2) SU(2)/ U(1) S 2 S 3 C(N 1,1 ) is the conifold {z1 2 + z2 2 + z2 3 + z2 4 = 0} C4 Up to symmetries the conifold admits two AC CY desingularisations (Candelas de la Ossa, 1990): the small resolution: total space of O( 1) O( 1) P 1 tip of the cone replaced by a round totally geodesic holomorphic S 2 the smoothing: T S 3 tip of the cone replaced by a round totally geodesic special Lagrangian S 3 The conifold itself and its asymptotically conical CY desingularisations are cohomogeneity one: SU(2) SU(2) acts isometrically with generic orbit of codimension one Question: Can we desingularise SC(N 1,1 ) as a cohomogeneity one nk space?

6 Cohomogeneity one nk manifolds 2010, Podestà Spiro: possible group diagrams of complete cohomogeneity one nk 6 manifolds G K K + K M SU(2) SU(2) U(1) SU(2) SU(2) S 3 S 3 SU(2) SU(2) U(1) SU(2) U(1) SU(2) S 6 SU(2) SU(2) U(1) U(1) SU(2) SU(2) U(1) CP 3 SU(2) SU(2) U(1) U(1) SU(2) U(1) SU(2) S 2 S 4 SU(3) SU(2) SU(3) SU(3) S 6

7 Outline of the proof 1. Understand the local theory for cohomogeneity one nk 6 manifolds in a neighbourhood of the principal orbit 2. Understand local cohomogeneity one nk manifolds in a neighbourhood of a singular orbit 3. Match pairs of solutions extending smoothly over a singular orbit

8 Nearly hypo structures On (0, T ) N 1,1 we write ω = η dt + ω 1 Ω = (ω 2 + iω 3 ) (η + idt), where (η, ω 1, ω 2, ω 3 ) defines an invariant SU(2) structure on N 1,1 The (NK) equation decouples into static and evolution equations for a family of invariant SU(2) structures on N 1,1 The SU(2) structure induced on a hypersurface in a nk 6 manifold is called a nearly hypo structure. It satisfies dω 1 = 3η ω 2 d(η ω 3 ) = 2ω 1 ω 1 The evolution equations to obtain a nk manifold by flowing a nearly hypo structure are: t ω 1 = dη 3ω 3 t (η ω 2 ) = dω 3 t (η ω 3 ) = dω 2 + 4η ω 1 Proposition The space of invariant nearly hypo structures on N 1,1 is a smooth manifold diffeomorphic to an open connected subset of SO 0 (1, 2). Up to symmetries, there exists a 2 parameter family of local cohomogeneity one nk stuctures on (0, T ) N 1,1.

9 Solutions extending over a singular orbit Problem: recognise which local solutions extend to complete metrics Proceed in two steps; separate the two singular orbits that appear and study separately. 1. Understand which solutions extend over a given singular orbit. 2. Understand how to match a pair of solutions from the previous step. Podestà Spiro: up to symmetries possible singular orbits are SU(2) SU(2)/U(1) SU(2) S 2 SU(2) SU(2)/ SU(2) S 3 Proposition There exist two 1 parameter families {Ψ a } a>0 and {Ψ b } b>0 of solutions to the fundamental ODE system which extend smoothly over a singular orbit S 2 and S 3, respectively. As a, b 0, appropriately rescaled the local nk structures Ψ a and Ψ b converge to the CY structure on the small resolution and the smoothing, respectively.

10 Maximal volume orbits M complete = orbital volume function V has a unique maximum The space of invariant maximal volume orbits V: V R 2 V 1 on V and V = 1 precisely for the Sasaki Einstein structure on N 1,1 V {V C} is compact Proposition Every member of the families {Ψ a } a>0 and {Ψ b } b>0 has a unique maximal volume orbit. Proof uses a simple continuity argument in the parameter a or b Non-emptiness: 3 of 4 known homogeneous examples appear in the families {Ψ a } a>0 and {Ψ b } b>0 Openness follows from the nondegeneracy of critical points of V Closedness uses compactness of V {V C} plus standard ODE theory and basic comparison theory

11 Matching Strategy for finding complete nk metrics: Match pairs of solutions in the two families across their maximal volume orbits using discrete symmetries α, β continuous curves in V R 2 parametrising the maximal volume orbits of {Ψ a } a>0 and {Ψ b } b>0 (u, v) : V R 2 coordinates on V Discrete symmetries = reflections along the axes Matching means: curves α and β must intersect or self-intersect (after applying a discrete symmetry) or intersect either axis An intersection point of α with axis u = 0 gives S 2 S 4, with axis v = 0 gives CP 3 An intersection point of β with either axis gives S 3 S 3 An intersection point of α with β (after the action of a discrete symmetry) gives S 6 Question: How many intersection/self-intersection points do the curves α and β (and their images under the reflections) have?

12 The curves α and β The homogeneous nk structures on CP 3, S 6 and S 3 S 3 give us 3 points on the curves α and β: CP 3, S 3 S 3 gives intersection points of α and β with the axes; S 6 gives an intersection point between α and (the reflection of) β. Study limits of α and β curves as the parameters a and b 0: Proposition As a, b 0 Ψ a and Ψ b converge to the sine-cone over the standard Sasaki Einstein structure on N 1,1. Proof uses: convergence after rescaling to AC CY structures the Böhm functional B for cohom 1 Einstein metrics scale-invariance of B and the fact that B = 20V 2 5 on a max vol orbit SE metric on N 1,1 is the absolute min of V on space V of max vol orbits Maximal volume orbit of SC(N 1,1 ) corresponds to origin in R 2 u,v so curves α and β come out of the origin

13 An inhomogeneous nk structure on S 3 S 3 Aim: find an intersection point of the curve β with the axis v = 0 C(b) = # zeroes of v before the max vol orbit of Ψ b (u, v) satisfies the system { u = 3 λ v v = 4λu, zeroes of v are non-degenerate = C(b) can only jump when zero of v occurs on max vol orbit b = 1 is standard S 3 S 3 and we check C(b) = 1 Show that C(b) 2 for all b sufficiently small. Convergence to sine-cone implies λ(t) sin t as b 0 Compare to a solution of the Legendre equation (sin t u ) + 12 sin t u = 0 Theorem There exists at leat one b (0, 1) such that v = 0 on the max vol orbit and thus at least one smooth inhomogeneous nk structure on S 3 S 3.

14 An inhomogeneous nk structure on S 6 Aim: force α and β to intersect in a second point besides the one giving the homogeneous nk structure on S 6 Idea: use the two solutions on S 3 S 3 to find a closed bounded region D in the plane encircling the origin; D is given by an arc of the curve β and its reflections along the axes The α curve starts at the origin (as a 0); want to show that α eventually leaves D Proposition The curve α exits any compact subset of R 2 as a. Idea of proof: based on explicit Taylor series for solution Ψ a consider a particular (non geometric) rescaling of Ψ a Show that solutions to the rescaled equations converge smoothly to some limiting object Scaling used shows V max ca 4

15 Summary Conjecture The Main Theorem yields all (inhomogeneous) cohomogeneity one nk structures on simply connected 6 manifolds α H S 6 std 0.4 CP 3 std w2 0.2 S 3 S 3 new S 6 new β H S 3 S 3 std

16 Deformation theory of nk manifolds Moroianu Nagy Semmelmann (2008): infinitesimal nk deformations modulo gauge identified with co-closed primitive (1, 1) forms of eigenvalue 12 for the Laplacian Moroianu Semmelmann (2010): all homogeneous nearly Kähler manifolds are rigid except for F 3 Theorem (F., 2016) All the infinitesimal deformations of F 3 are obstructed Proof uses Hitchin s volume functionals to reinterpret NK structures as the zeroes of a smooth map Φ : Ω 3 exact Ω 4 exact Ω 1 Ω 3 exact Ω 4 exact Dirac-type operators on nk manifolds are used to show that the linearisation of Φ together with a gauge fixing condition is an elliptic linear PDE with kernel and cokernel equal to Ω 1,1 0 (12) Non-vanishing obstructions to lift the infinitesimal deformations of F 3 to second order deformations (cf. Koiso (1982) for the Einstein case)

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