Almost holomorphic curves in real analytic hypersurfaces
Pierre Bonneau, Emmanuel Mazzilli
TL;DR
This paper develops an exterior differential systems (EDS) framework to study germs of pseudo-holomorphic disks contained in real analytic hypersurfaces within almost complex manifolds endowed with a real analytic $J$. By formulating the existence problem as a Pfaffian system and examining torsion and tableau dimensions (and their prolongations), the authors connect solvability to involution in the Cartan sense and to Levi-form–type obstructions. In the almost complex case they show that a key determinant, $\mathfrak{D_0}$, vanishes, enabling torsion absorption under suitable linear relations, while in the complex case explicit Levi form criteria emerge, constraining possible discs. The paper also proposes a practical stratified test that reduces the problem to a finite family of torsion-free systems on analytic strata and provides examples illustrating when holomorphic discs exist or are obstructed, shedding light on the geometric structure of pseudoholomorphic foliations on real analytic hypersurfaces.
Abstract
Using the theory of exterior differential systems, we study the existence of germ of pseudo-holomorphic disk in a real analytic hypersurface locally defined in a complex manifold equipped with J a real analytic almost complex structure. The integrable case in C n with J the multiplication by i has been intensively studied by several authors [DF], [DA1] and [DA2] for example. The non integrable case is drastically different essentially due to the following fact : in generic case, there is no J-invariant objects of dimension bigger than one. This simple observation leads to the non existence of some equivalents of Segree varieties or ideals of holomorphic functions which play a fundamental role in the complex case. Nevertheless in the almost complex case, we adopt the exterior differential system point of view of E.Cartan developed and clarified in [BCGGG].