Equivalences of 5 -Dimensional CR Manifolds [Joint Work with Wei-Guo - PowerPoint PPT Presentation

Equivalences of 5 -Dimensional CR Manifolds [Joint Work with Wei-Guo Foo and The-Anh Ta] Einstein-Weyl Structures [Joint Work with Paweł Nurowski] Differential Invariants of Parabolic Surfaces [Joint Work with Zhangchi Chen] J OËL M ERKER www.math.u-psud.fr/ ∼ merker/ Département de Mathématiques d’Orsay Bibliothèque mathématique Jacques Hadamard 2019 Taipei Conference on Complex Geometry Institute of Mathematics, Academia Sinica Wednesday 18 December 2019

� � � 2 Cartan’s Method of Equivalence Cartan devised a quite sophisticated and proteiform method of equiva- lence . Given a manifold M equipped with a certain class of geometric, say CR here, structures, Cartan’s method of equivalence consists in constructing a bundle π : P − → M together with an absolute (co)parallelism on P , namely a coframe of everywhere linearly independent 1 -forms θ 1 , . . . , θ dim P on P such that: Π P ′ P π ′ π � M ′ M Φ → M ′ between two CR manifolds • every local CR diffeomorphism Φ: M − → P ′ satisfying Π ∗ θ ′ i = θ i for lifts uniquely as a diffeomorphism Π: P − 1 � i � dim P , with P ′ and the θ ′ i similarly constructed; → P ′ commuting with projec- • conversely, every diffeomorphism Π: P − tions π , π ′ whose horizontal part is a diffeomorphims M − → M ′ and which

satisfies Π ∗ θ ′ i = θ i for 1 � i � dim P , has a horizontal part which is Cauchy- Riemann diffeomorphism (or, more generally, a diffeomorphism respecting the considered geometric structure). [Beyond, there can exist Cartan connections associated to (modifications of) P − → M , but we will not need this concept.] Rexpressing the exterior differentials dθ i and dθ ′ i from both sides in terms of the basic 2 -forms provided by the two ambient coframes: � � dθ ′ i = T ′ i j,k ( p ′ ) θ ′ j ∧ θ ′ k , dθ i = T i j,k ( p ) θ j ∧ θ k and j<k j<k certain structure functions appear, defined for p ∈ P and for p ′ ∈ P ′ , and the exact pullback relations Π ∗ θ ′ i = θ i force individual invariancy of all them: � � T ′ i = T i Φ( p ) j,k ( p ) ( ∀ p ∈ P ) . j,k As is known, Cartan’s method is computationally extremely intensive , es- pecially in CR geometry, where several normalizations and prolongations are required. Explicit expressions of intermediate torsion coefficients which con- duct to the final T i j,k ( p ) grow dramatically in complexity.

One reason for such a complexity is the presence of large isotropy groups 4 for the CR automorphisms groups of (standard) models, which imposes a great number of steps. Another reason is the nonlinear character of differential al- gebraic polynomial expressions that must be handled progressively. The last reason is that Cartan’s method studies geometric structures at every point of the base manifold, and there is a price to pay for this generality. In most existing references ( cf. the bibliography), the trick that Cartan him- self devised to avoid nonlinear complications while retaining anyway some es- sential information, is the so-called Cartan Lemma . It is explicit only at the level of linear algebra. Even admitting to only deal with linear algebra com- putations, as Chern always did, Cartan’s method is often long and demanding.

2 -Nondegenerate Levi Rank 1 Hypersurfaces M 5 ⊂ C 3 [Joint Work with Wei-Guo Foo and The-Anh Ta] • Coordinates: � � ∈ C 3 . z, ζ, w The right graphed equation for the model light cone M LC ⊂ C 3 in C 2 , 1 was discovered by Gaussier-M. in 2003: u = zz + 1 2 z 2 ζ + 1 2 z 2 ζ � � M LC : =: ♠ z, ζ, z, ζ , 1 − ζζ Start with M 5 ⊂ C 3 , with 0 ∈ M , rigid, graphed as: u = F ( z, ζ, z, ζ ) . Constant Levi rank 1 means, possibly after a linear transformation in C 2 z,ζ , that: � � � � F zz F ζz � � (0.1) F zz � = 0 ≡ � =: Levi ( F ) , � � F zζ F ζζ �

while 2 -nondegeneracy means that: 6 � � � � F zz F zz � � (0.2) 0 � = � . � � F zzζ F zzζ � At the origin, M LC of equation: 2 z 2 ζ + 1 2 z 2 ζ + O z,ζ,z,ζ (4) , u = zz + 1 is obviously 2 -nondegenerate, thanks to the cubic monomial 1 2 z 2 ζ which gives � � � 1 0 � = 1 . As for constant Levi rank 1 , that (0.2) at ( z, ζ ) = (0 , 0) becomes ∗ 1 order two terms u = zz + · · · show that this condition is true at the origin, and simple computations show that (0.1) is identically zero: � � � � z + zζ 1 � � � � (1 − ζζ ) 2 1 − ζζ ≡ 0 ( – indeed! ) . � � ( z + zζ )( z + zζ ) z + zζ � � � � (1 − ζζ ) 2 (1 − ζζ ) 3 Consider as before a rigid M 5 ⊂ C 3 with 0 ∈ M , which is 2 -nondegenerate and has Levi form of constant rank 1 , i.e. belongs to the class C 2 , 1 , and which is graphed as: � � u = F z 1 , z 2 , z 1 , z 2 .

The letter ζ is protected, hence not used instead of z 2 , since ζ will denote a 1 -form. The two natural generators of T 1 , 0 M and T 0 , 1 M are: and L 1 := ∂ z 1 − i F z 1 ∂ v L 2 := ∂ z 2 − i F z 2 ∂ v , in the intrinsic coordinates ( z 1 , z 2 , z 1 , z 2 , v ) on M . The Levi kernel bundle K 1 , 0 M ⊂ T 1 , 0 M is generated by: ❦ := − F z 2 z 1 where K := ❦ L 1 + L 2 , , F z 1 z 1 is the slant function. The hypothesis of 2 -nondegeneracy is equivalent to the nonvanishing: 0 � = L 1 ( ❦ ) . Also, the conjugate K generates the conjugate Levi kernel bundle K 0 , 1 ⊂ T 0 , 1 M . There is a second fundamental function, and no more: P := F z 1 z 1 z 1 . F z 1 z 1 In the rigid case, it looks so simple ! But in the nonrigid case, P has a numerator involving 69 differential monomials !

Foo-Merker-Ta produced reduction to an { e } -structure for the equivalence 8 problem, under rigid (local) biholomorphic transformations, of such rigid M 5 ∈ C 2 , 1 . Theorem. [Foo-M.-Ta 2019] There exists an invariant 7 -dimensional bundle P 7 − → M 5 equipped with coordinates: � � z 1 , z 2 , z 1 , z 2 , v, c , c , with c ∈ C , together with a collection of seven complex-valued 1 -form which make a frame for TP 7 , denoted: � � ρ, κ, ζ, κ, ζ, α, α ( ρ = ρ ) , which satisfy 7 invariant structure equations of the form: � � dρ = α + α ∧ ρ + i κ ∧ κ, dκ = α ∧ κ + ζ ∧ κ, � � ∧ ζ + 1 c ■ 0 κ ∧ ζ + 1 dζ = α − α cc ❱ 0 κ ∧ κ, dα = ζ ∧ ζ − 1 c ■ 0 ζ ∧ κ + 1 cc ◗ 0 κ ∧ κ + 1 c ■ 0 ζ ∧ κ, conjugate structure equations for dκ , dζ , dα being easily deduced.

Here, as in Pocchiola’s Ph.D., there are exactly two primary Cartan- curvature invariants: � � �� � � � � L 1 ( ❦ ) L 1 ( ❦ ) L 1 ( ❦ ) K L 1 K L 1 ■ 0 := − 1 + 1 + L 1 ( ❦ ) 2 L 1 ( ❦ ) 3 3 3 � � � � L 1 ( ❦ ) L 1 ( ❦ ) L 1 L 1 + 2 + 2 , 3 3 L 1 ( ❦ ) L 1 ( ❦ ) � � �� � � � L 1 � 2 L 1 L 1 L 1 ( ❦ ) L 1 ( ❦ ) ❱ 0 := − 1 + 5 − 3 9 L 1 ( ❦ ) L 1 ( ❦ ) � � L 1 ( ❦ ) L 1 − 1 P + 1 3 L 1 ( P ) − 1 9 P P . 9 L 1 ( ❦ ) One can check that Pocchiola’s ❲ 0 which occurs under general biholomorphic transformations of C 3 (not necessarily rigid!), when written for a rigid M 5 ⊂ C 3 , identifies with: � � � � F ( z 1 , z 2 , z 1 , z 2 ) ≡ ❲ 0 F ( z 1 , z 2 , z 1 , z 2 ) . ■ 0

Furthermore, there is one secondary invariant whose unpolished expression 10 is: � � � � � � � � � � L 1 ( ❦ ) L 1 ( ❦ ) P − L 1 P − L 1 ◗ 0 := 1 − 1 ■ 0 − 1 ■ 0 − 1 K ( ❱ 0 ) 2 L 1 L 1 ( ❦ ) . ■ 0 3 6 2 L 1 ( ❦ ) L 1 ( ❦ ) Visibly indeed, the vanishing of ■ 0 and ❱ 0 implies the vanishing of ◗ 0 . In fact, a consequence of Cartan’s general theory is: M is rigidly equivalent to the Gaussier-Merker model . 0 ≡ ■ 0 ≡ ❱ 0 ⇐ ⇒ By deducing new relations from the structure equations above, it was proved that ◗ 0 is real-valued, but a finalized expression was missing there. A clean finalized expression of ◗ 0 , in terms of only the two fundamental

functions ❦ , P (and their conjugates), from which one immediately sees real- valuedness, is: � � � � 2 � 1 K L 1 ( ❦ ) L 1 L 1 ( ❦ ) ◗ 0 := 2 Re − L 1 ( ❦ ) 4 9 � � �� � � � � � � K L 1 L 1 ( ❦ ) L 1 L 1 ( ❦ ) K L 1 ( ❦ ) L 1 L 1 ( ❦ ) − 1 − 1 P − L 1 ( ❦ ) 3 L 1 ( ❦ ) 3 9 9 � � � � � � �� L 1 L 1 ( ❦ ) L 1 L 1 ( ❦ ) K L 1 L 1 ( ❦ ) − 1 + 1 P − L 1 ( ❦ ) 2 L 1 ( ❦ ) 2 9 9 � � � � � � �� L 1 ( ❦ ) L 1 ( ❦ ) L 1 ( ❦ ) L 1 L 1 L 1 L 1 − 2 P − 1 P + 1 + 1 6 L 1 ( P 9 9 3 L 1 ( ❦ ) L 1 ( ❦ ) L 1 ( ❦ ) � � � � 2 � � � � L 1 L 1 ( ❦ ) − 1 � 2 + 1 � P � � . � � 9 3 L 1 ( ❦ )