Toric Nearly K ahler 6-manifolds Andrei Moroianu CNRS - Paris-Sud - PDF document

Toric Nearly K¨ ahler 6-manifolds Andrei Moroianu CNRS - Paris-Sud University Progress and Open Problems 2019: September 8-11, 2019, SCGP, Stony Brook – joint work with Paul-Andi Nagy –

Nearly K¨ ahler manifolds were originally in- troduced as the class W 1 in the Gray-Hervella classification of almost Hermitian manifolds. More precisely, an almost Hermitian manifold ( M 2 n , g, J ) is called nearly K¨ ahler (NK) if ( ∇ X J )( X ) = 0 for every vector field X on M , where ∇ denotes the Levi-Civita covariant derivative of g . A NK manifold is called strict if ( ∇ J ) p � = 0 for every p ∈ M . Remark. In dimension 2 n = 4, NK = K¨ ahler.

Examples: • K¨ ahler manifolds. • twistor spaces over positive QK manifolds, endowed with the non-integrable almost com- plex structure and with the metric rescaled by a factor 2 on the fibres. • naturally reductive 3-symmetric spaces G/H where G is compact, H is the invariant group of an automorphism σ of G of or- der 3, g = h ⊕ p , and p has a scalar product such that for every X, Y, Z ∈ p : � [ X, Y ] p , Z � + � [ X, Z ] p , Y � = 0 . The almost complex structure is determined by the endomorphism J of p satisfying √ σ ∗ = − 1 3 2Id p + 2 J.

A product of NK manifolds is again NK. Con- versely, the factors of the de Rham decompo- sition of a NK manifold are NK. Theorem. (Nagy 2002): Every simply con- nected, complete, de Rham irreducible NK man- ifold is either one of the above examples, or a strict NK 6-manifold. From now on, we restrict our attention to strict NK 6-manifolds. These are interesting for several reasons:

Properties of strict NK 6-manifolds: • carry real Killing spinors positive Ein- � stein; after rescaling the metric, one can normalize them to having scalar curvature 30 (like the round S 6 ). • ∇ J has constant norm SU(3)-structure � • carry a connection with parallel and skew- symmetric torsion ∇ X = ∇ X − 1 ˜ 2 J ◦ ∇ X J • the Riemannian cone ( M × R ∗ + , t 2 g +d t 2 ) of a normalized NK 6-manifold ( M, g, ω ) has holonomy contained in G 2 , defined by the positive 3-form ϕ = 1 3d( t 3 ω ) = 1 3 t 3 d ω + t 2 d t ∧ ω

Main problem: lack of examples. 3-symmetric spaces were classified by Gray. In dimension 6: • S 6 = G 2 / SU(3) • SU(2) × SU(2) × SU(2) / ∆ ∼ S 3 × S 3 • Sp(2) / U(2) ∼ C P 3 • SU(3) / U(1) × U(1) ∼ F (1 , 2). (Butruille 2004) These are all Theorem. homogeneous SNK 6-manifolds.

Foscolo and Haskins (2017): 2 new examples (of cohomogeneity 1) on S 6 and S 3 × S 3 , both with isometry group SU(2) × SU(2). Deformations of SNK 6-manifolds were stud- ied by –, Nagy, Semmelmann (2008, 2010, 2011). The moduli space is isomorphic to the space of co-closed primitive (1 , 1)-forms which are eigenforms of the Laplace operator for the eigen- value 12. Using representation theory one can com- pute this space on the homogeneous exam- ples. It vanishes except on F (1 , 2) where it has dimension 8. However, these infinitesimal deformations are obstructed (Foscolo 2017).

SU(3)-structures on SNK 6-manifolds Let M 6 be an oriented manifold. An SU(3)- structure on M is a triple ( g, J, ψ ), where • g is a Riemannian metric, • J is a compatible almost complex structure (i.e. ω := g ( J · , · ) is a 2-form), • ψ = ψ + + iψ − is a (3 , 0) complex volume form satisfying ψ + ∧ ψ − = 4vol g = 2 3 ω 3 . It is possible to characterize SU(3)-structures in terms of exterior forms only (Hitchin).

Lemma 1 A pair ( ω, ψ + ) ∈ C ∞ (Λ 2 M × Λ 3 M ) defines an SU(3) -structure on M provided that: • ω 3 � = 0 (i.e. ω is non-degenerate). • ω ∧ ψ + = 0 . • If K ∈ End(TM) ⊗ Λ 6 M is defined by K ( X ) := ( X � ψ + ) ∧ ψ + ∈ Λ 5 M ≃ TM ⊗ Λ 6 M , then tr K 2 = − 1 6 ( ω 3 ) 2 ∈ (Λ 6 M ) ⊗ 2 • ω ( X, K ( X )) /ω 3 > 0 for every X � = 0 . “Proof”: Define J := 6 K/ω 3 , g ( · , · ) := ω ( · , J · ), ψ − := − ψ + ( J · , · , · ).

A normalized SNK structure ( g, J, ω ) on M 6 SU(3)-structure ( g, J, ω, ψ + , ψ − ) where � ψ + := ∇ ω, ψ − := − ψ + ( J · , · , · ) . This satisfies the exterior differential system  d ω = 3 ψ +  d ψ − = − 2 ω 2 .  Conversely, an SU(3)-structure satisfying this system is a normalized SNK structure (Hitchin). This is similar to the case of G 2 structures, where a stable 3-form is parallel if and only if it is harmonic.

Toric NK 6-manifolds An infinitesimal automorphism of a normal- ized SNK 6-manifold ( M, g, J, ω, ψ ± ) is a vector field ξ whose flow preserves the whole struc- ture (enough to have L ξ ω = 0 = L ξ ψ + ). Lemma. rk( aut ( M, g, J )) ≤ 3. If equality holds, ( M, g, J ) is called toric. The only homogeneous example is S 3 × S 3 . Assume that ( M, g, J ) is toric and let ξ 1 , ξ 2 , ξ 3 be a basis of a Cartan subalgebra of aut ( M, g, J ). Lemma. The vector fields ξ 1 , ξ 2 , ξ 3 , Jξ 1 , Jξ 2 , Jξ 3 are linearly independent on a dense open subset M 0 of M . dual basis { θ 1 , θ 2 , θ 3 , γ 1 , γ 2 , γ 3 } of Λ 1 M 0 . �

Define the functions ε := ψ − ( ξ 1 , ξ 2 , ξ 3 ) . µ ij := ω ( ξ i , ξ j ) ,  d ω = 3 ψ +  The Cartan formula and � d ψ − = − 2 ω 2  d µ ij = d( ξ j � ξ i � ω ) = − ξ j � d( ξ i � ω ) ξ j � ξ i � d ω = − 3 ξ i � ξ j � ψ + . = Similarly, d( ξ 3 � ξ 2 � ξ 1 � ψ − ) = − ξ 3 � ξ 2 � ξ 1 � d ψ − d ε = 2 ξ 3 � ξ 2 � ξ 1 � ω 2 . = Remarks: 1. ψ + ( ξ 1 , ξ 2 , ξ 3 ) = 0 on M . 2. ε does not vanish on M 0 .

It follows that the map µ : M → Λ 2 R 3 ∼ = so (3) defined by   0 µ 12 µ 13 µ := µ 21 0 µ 23     µ 31 µ 32 0 is the multi-moment map of the strong geom- etry ( M, ψ + ) defined by Madsen and Swann (and studied further by Dixon in the particular case where M = S 3 × S 3 ). Similarly, the function ε is the multi-moment map associated to the stable closed 4-form d ψ − .

Consider the symmetric 3 × 3 matrix C := ( C ij ) = ( g ( ξ i , ξ j )) . In terms of the basis { θ 1 , θ 2 , θ 3 , γ 1 , γ 2 , γ 3 } of Λ 1 M 0 we can write ψ + = ε ( γ 123 − θ 12 ∧ γ 3 − θ 31 ∧ γ 2 − θ 23 ∧ γ 1 ) , ψ − = ε ( θ 123 − γ 12 ∧ θ 3 − γ 31 ∧ θ 2 − γ 23 ∧ θ 1 ) , where γ 123 = γ 1 ∧ γ 2 ∧ γ 3 etc. Similarly, 3 µ ij ( θ ij + γ ij ) + C ij θ i ∧ γ j � � ω = 1 ≤ i<j ≤ 3 i,j =1 The normalization condition ψ + ∧ ψ − = 2 3 ω 3 translates into 3 det( C ) = ε 2 + � C ij y i y j , i,j =1 where y 1 := µ 23 , y 2 := µ 31 , y 3 := µ 12 .

The previous formula d µ ij = − 3 ξ i � ξ j � ψ + can be restated as d y i = − 3 εγ i , i = 1 , 2 , 3 . Similarly, d ε = 2 ξ 3 � ξ 2 � ξ 1 � ω 2 is equivalent to 3 C ij y i γ j . � d ε = 4 i,j =1 Remark also that ξ j � d θ i = 0 explicit � expression of d θ i in terms of γ j , y j , ε and C . Let U := M 0 /T 3 be the set of orbits of the T 3 -action generated by the vector fields ξ i . All invariant functions and basic forms de- y i , ε , γ i , C ij , etc. scend to U Since ε � does not vanish on M 0 { y i } define a lo- � cal coordinate system on U .  d ω = 3 ψ +  The system Key point: d ψ − = − 2 ω 2  ∃ ϕ on U such that Hess( ϕ ) = C in the � coordinates { y i } .

Let us introduce the operator ∂ r of radial differentiation, acting on functions on U by 3 ∂f � ∂ r f := y i . ∂y i i =1 The function ϕ can be chosen in Claim: such a way that ε 2 = 8 3( ϕ − ∂ r ϕ ) . Proof: It is enough to show that the exterior derivatives of the two terms coincide. Since 3 ∂ 2 ϕ ∂ ( ∂ r ϕ ) y i + ∂ϕ � = , ∂y j ∂y i ∂y j ∂y j i =1 we get: 3 3 C ij y i εγ j � � d( ∂ r ϕ − ϕ ) = C ij y i d y j = − 3 i,j =1 i,j =1 − 3 4 ε d ε = − 3 8d( ε 2 ) . =

On the other hand, 3 3 ∂f ∂ 2 � � r ϕ = ∂ r ( y i ) = C ij y i y j + ∂ r ϕ. ∂y i i =1 i,j =1 Summing up, the previous relation 3 det( C ) = ε 2 + � C ij y i y j i,j =1 becomes: det(Hess( ϕ )) = 8 3 ϕ − 11 3 ∂ r ϕ + ∂ 2 r ϕ. This Monge-Amp` ere equation is enough to recover (locally) the full structure of the toric SNK manifold provided some positivity con- straints hold.

The inverse construction We will show that a solution ϕ of det(Hess( ϕ )) = 8 3 ϕ − 11 3 ∂ r ϕ + ∂ 2 r ϕ on some open set U ⊂ R 3 defines a toric SNK structure on U 0 × T 3 , where U 0 is some open subset of U . Let y 1 , y 2 , y 3 be the standard coordinates on U and let µ be the 3 × 3 skew-symmetric matrix   0 y 3 − y 2 µ := 0 − y 3 y 1  .    0 y 2 − y 1 Define the 6 × 6 symmetric matrix � � Hess( ϕ ) − µ D := . µ Hess( ϕ ) Let U 0 ⊂ U denote the open subset U 0 := { x ∈ U | ϕ ( x ) − ∂ r ϕ ( x ) > 0 and D > 0 } .