A geometric approach to Catlin’s boundary systems

Zaitsev, Dmitri

doi:10.5802/aif.3304

A geometric approach to Catlin’s boundary systems [ Une approche géométrique aux systèmes bord de Catlin ]

Zaitsev, Dmitri

Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2635-2679.

Résumé
Abstract

Soit $p$ un point d’une hypersurface $M \subset ℂ^{n},$ réelle et lisse, en lequel la forme de Levi possède un noyau non trivial, noté $K_{p}^{10} \neq 0$ . On introduit alors un tenseur d’ordre $3,$ invariant

τ_{p}^{3} : ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

qui, avec le tenseur d’ordre $3$ d’Ebenfelt, fournit l’ensemble de tous les tenseurs d’ordre $3$ invariants de $M$ au point $p$ .

Si, de plus, on suppose que $M$ est pseudo-convexe, alors $τ_{p}^{3}$ est identiquement nul. On définit alors un tenseur d’ordre $4$ , invariant

τ_{p}^{4} : ℂ T_{p} \times ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}) .

On introduit également pour $q = 0, ..., n - 1$ , un faisceau de sous-modules invariant, noté $𝒮^{10} (q)$ , constitué de champs de vecteurs de type $(1, 0)$ , construits à partir de la forme de Levi, ainsi qu’un faisceau d’idéaux invariant, noté $ℐ (q)$ , constitué de fonctions complexes construites à partir de certaines composantes et dérivées de la forme de Levi. L’ensemble des points dont la forme de Levi est de rang $q$ est alors contenu localement dans des sous-variétés réelles et lisses définies par des parties rélles de fonctions dans $ℐ (q)$ , et dont les espaces tangents ont des descriptions algébriques explicites en termes du tenseur $τ^{4}$ .

Tout récemment, la construction de $τ^{3}$ et $τ^{4}$ a inspiré Raich et Harrington [35] qui ont découvert de nouveaux invariants associés à ces derniers dans le cas où $M$ n’est pas pseudo-convexe et les ont reliés à la « closed range property » pour l’opérateur $\bar{\partial}$ .

Enfin, on fait le lien entre ces invariants définis plus haut et le type fini au sens de D’Angelo, le multitype au sens de Catlin et les « boundary systems » de Catlin.

For a point $p$ in a smooth real hypersurface $M \subset ℂ^{n}$ , where the Levi form has the nontrivial kernel $K_{p}^{10} \neq 0$ , we introduce an invariant cubic tensor

τ_{p}^{3} : ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

which together with Ebenfelt’s $3$ rd order tensor, constitutes the full set of the $3$ rd order invariants of $M$ at $p$ .

Next, in addition, assume $M \subset ℂ^{n}$ to be (weakly) pseudoconvex. Then $τ_{p}^{3}$ must identically vanish. In this case we further define an invariant quartic tensor

τ_{p}^{4} : ℂ T_{p} \times ℂ T_{p} \times K_{p}^{10} \times \bar{K_{p}^{10}} \to ℂ \otimes (T_{p} / H_{p}),

and for every $q = 0, \dots, n - 1$ , an invariant submodule sheaf $𝒮^{10} (q)$ of $(1, 0)$ vector fields in terms of the Levi form, and an invariant ideal sheaf $ℐ (q)$ of complex functions generated by certain components and derivatives of the Levi form, such that the set of points of Levi rank $q$ is locally contained in real smooth submanifolds defined by real parts of the functions in $ℐ (q)$ , whose tangent spaces have explicit algebraic description in terms of the quartic tensor $τ^{4}$ .

Most recently, the constructions of $τ^{3}$ and $τ^{4}$ inspired Raich and Harrington [35] to discover new related invariants in the non-pseudoconvex case and connect them with the closed range property for the $\bar{\partial}$ operator.

Finally, we relate the introduced invariants with D’Angelo’s finite type, Catlin’s multitype and Catlin’s boundary systems.

Reçu le : 2018-01-21
Accepté le : 2018-07-11
Publié le : 2019-10-29

DOI : https://doi.org/10.5802/aif.3304
Classification : 32T25, 32T27, 32V05, 32V15, 32V35, 32W05, 32S60, 58K50
Mots clés: multitype de Catlin, estimée sous-elliptique, système bord, forme de Levi, pseudo-convexité, hypersurface réelle, tenseur invariant, faisceau idéal

@article{AIF_2019__69_6_2635_0,
     author = {Zaitsev, Dmitri},
     title = {A geometric approach to Catlin's boundary systems},
     journal = {Annales de l'Institut Fourier},
     pages = {2635--2679},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     doi = {10.5802/aif.3304},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2019__69_6_2635_0/}
}

Zaitsev, Dmitri. A geometric approach to Catlin’s boundary systems. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2635-2679. doi : 10.5802/aif.3304. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2635_0/

Bibliographie

[1] Baracco, Luca; Khanh, Tran Vu; Pinton, Stefano; Zampieri, Giuseppe Hölder regularity of the solution to the complex Monge-Ampère equation with $L^{p}$ density, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 4, 74, 8 pages | Zbl 1361.32045

[2] Bharali, Gautam; Stensønes, Berit Plurisubharmonic polynomials and bumping, Math. Z., Volume 261 (2009) no. 1, pp. 39-63 | Article | MR 2452636 | Zbl 1185.32025

[3] Biard, Séverine; Straube, Emil J. $L^{2}$ -Sobolev theory for the complex Green operator, Int. J. Math., Volume 28 (2017) no. 9, 1740006, 31 pages | MR 3690415 | Zbl 1380.32036

[4] Bierstone, Edward; Milman, Pierre D. Ideals of holomorphic functions with $C^{\infty}$ boundary values on a pseudoconvex domain, Trans. Am. Math. Soc., Volume 304 (1987) no. 1, pp. 323-342 | MR 906818 | Zbl 0631.32015

[5] Boas, Harold P. The Szegö projection: Sobolev estimates in regular domains, Trans. Am. Math. Soc., Volume 300 (1987) no. 1, pp. 109-132 | Zbl 0622.32006

[6] Boas, Harold P.; Straube, Emil J.; Yu, Ji-Ye Boundary limits of the Bergman kernel and metric, Mich. Math. J., Volume 42 (1995) no. 3, pp. 449-461 | MR 1357618 | Zbl 0853.32028

[7] Catlin, David W. Global regularity of the $\bar{\partial}$ -Neumann problem, Complex analysis of several variables (Madison,1982) (Proceedings of Symposia in Pure Mathematics) Volume 41, American Mathematical Society, 1982, pp. 39-49 | Article | MR 740870 | Zbl 0578.32031

[8] Catlin, David W. Boundary invariants of pseudoconvex domains, Ann. Math., Volume 120 (1984), pp. 529-586 | Article | MR 769163 | Zbl 0583.32048

[9] Catlin, David W. Subelliptic estimates for the $\bar{\partial}$ -Neumann problem on pseudoconvex domains, Ann. Math., Volume 126 (1987), pp. 131-191 | Article | MR 898054 | Zbl 0627.32013

[10] Catlin, David W. Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z., Volume 200 (1989) no. 3, pp. 429-466 | Article | MR 978601 | Zbl 0661.32030

[11] Catlin, David W.; D’Angelo, John P. Subelliptic estimates, Complex analysis. Several complex variables and connections with PDE theory and geometry (Trends in Mathematics), Birkhäuser/Springer, 2010, pp. 75-94 | Zbl 1202.32027

[12] Çelik, Mehmet; Şahutoğlu, Sönmez On compactness of the $\bar{\partial}$ -Neumann problem and Hankel operators, Proc. Am. Math. Soc., Volume 140 (2012) no. 1, pp. 153-159 | Article | MR 2833527 | Zbl 1251.32032

[13] Charpentier, Philippe; Dupain, Yves Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat., Barc., Volume 50 (2006) no. 2, pp. 413-446 | Article | Zbl 1120.32002

[14] Charpentier, Philippe; Dupain, Yves Geometry of pseudo-convex domains of finite type with locally diagonalizable Levi form and Bergman kernel, J. Math. Pures Appl., Volume 85 (2006) no. 1, pp. 71-118 | Article | MR 2200592 | Zbl 1092.32017

[15] Charpentier, Philippe; Dupain, Yves Extremal bases, geometrically separated domains and applications, Algebra Anal., Volume 26 (2014) no. 1, pp. 196-269 (translation in St. Petersbg. Math. J. 26 (2015), no. 1, p. 139-191) | MR 3234809 | Zbl 1343.32023

[16] Chen, Bo-Yong; Fu, Siqi Comparison of the Bergman and Szegö kernels, Adv. Math., Volume 228 (2011) no. 4, pp. 2366-2384 | Article | Zbl 1229.32008

[17] Cho, Sanghyun A lower bound on the Kobayashi metric near a point of finite type in $ℂ^{n}$ , J. Geom. Anal., Volume 2 (1992) no. 4, pp. 317-325 | MR 1170478 | Zbl 0756.32015

[18] Cho, Sanghyun Boundary behavior of the Bergman kernel function on some pseudoconvex domains in $ℂ^{n}$ , Trans. Am. Math. Soc., Volume 345 (1994) no. 2, pp. 803-817 | MR 1254189 | Zbl 0813.32023

[19] Cho, Sanghyun Estimates of invariant metrics on pseudoconvex domains with comparable Levi form, J. Math. Kyoto Univ., Volume 42 (2002) no. 2, pp. 337-349 | MR 1966842 | Zbl 1036.32013

[20] Christ, Michael; Fu, Siqi Compactness in the $\bar{\partial}$ -Neumann problem, magnetic Schrödinger operators, and the Aharonov-Bohm effect, Adv. Math., Volume 197 (2005) no. 1, pp. 1-40 | Article | Zbl 1098.32020

[21] D’Angelo, John P. Subelliptic estimates and failure of semicontinuity for orders of contact, Duke Math. J., Volume 47 (1980) no. 4, pp. 955-957 | MR 596122 | Zbl 0455.32010

[22] D’Angelo, John P. Real hypersurfaces, orders of contact, and applications, Ann. Math., Volume 115 (1982), pp. 615-637 | Article | MR 657241 | Zbl 0488.32008

[23] D’Angelo, John P. Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, 1993 | Zbl 0854.32001

[24] Diederich, Klas; Herbort, Gregor Pseudoconvex domains of semiregular type, Contributions to complex analysis and analytic geometry (Aspects of Mathematics) Volume E26, Vieweg & Sohn, 1994, pp. 127-161 | Zbl 0845.32019

[25] Ebenfelt, Peter New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy, J. Differ. Geom., Volume 50 (1998) no. 2, pp. 207-247 | Article | MR 1684982 | Zbl 0945.32020

[26] Fornæss, John E.; McNeal, Jeffery D. A construction of peak functions on some finite type domains, Am. J. Math., Volume 116 (1994) no. 3, pp. 737-755 | Article | MR 1277453 | Zbl 0809.32005

[27] Fornæss, John E.; Sibony, Nessim Construction of P.S.H. functions on weakly pseudoconvex domains, Duke Math. J., Volume 58 (1989) no. 3, pp. 633-655 | MR 1016439 | Zbl 0679.32017

[28] Freeman, Michael Local biholomorphic straightening of real submanifolds, Ann. Math., Volume 106 (1977) no. 2, pp. 319-352 | Article | MR 463480 | Zbl 0372.32005

[29] Fu, Siqi; Jacobowitz, Howard The $\bar{\partial}$ -cohomology groups, holomorphic Morse inequalities, and finite type conditions, Pure Appl. Math. Q., Volume 6 (2010) no. 3, pp. 875-914 | MR 2677317 | Zbl 1214.32016

[30] Fu, Siqi; Straube, Emil J. Compactness in the $\bar{\partial}$ -Neumann problem, Complex analysis and geometry (Columbus, 1999) (Ohio State University Mathematical Research Institute Publications) Volume 9, Walter de Gruyter, 2001, pp. 141-160 | MR 1912737 | Zbl 1011.32025

[31] Ha, Ly Kim; Khanh, Tran Vu Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type, Math. Res. Lett., Volume 22 (2015) no. 2, pp. 467-484 | Zbl 1343.32030

[32] Harrington, Phillip S. Compact and subelliptic estimates for the $\bar{\partial}$ -Neumann operator on $C^{2}$ pseudoconvex domains, Math. Ann., Volume 337 (2007) no. 2, pp. 335-352 | Article | MR 2262787 | Zbl 1118.32027

[33] Harrington, Phillip S. Property (P) and Stein neighborhood bases on $C^{1}$ domains, Ill. J. Math., Volume 52 (2008) no. 1, pp. 145-151 | Article | MR 2507238 | Zbl 1175.32017

[34] Harrington, Phillip S. Global regularity for the $\bar{\partial}$ -Neumann operator and bounded plurisubharmonic exhaustion functions, Adv. Math., Volume 228 (2011) no. 4, pp. 2522-2551 | Article | MR 2836129 | Zbl 1328.32030

[35] Harrington, Phillip S.; Raich, Andrew Boundary invariants and the closed range property for $\bar{\partial}$ (2018) (https://arxiv.org/abs/1805.05793) | Zbl 06817615

[36] Henkin, Gennadi M.; Iordan, Andrei Compactness of the Neumann operator for hyperconvex domains with non-smooth B-regular boundary, Math. Ann., Volume 307 (1997) no. 1, pp. 151-168 | Article | MR 1427681 | Zbl 0869.32009

[37] Herbig, Anne-Katrin A sufficient condition for subellipticity of the $\bar{\partial}$ -Neumann operator, J. Funct. Anal., Volume 242 (2007) no. 2, pp. 337-362 | Article | MR 2274813 | Zbl 1118.32028

[38] Herbort, Gregor On the Bergman metric on bounded pseudoconvex domains an approach without the Neumann operator, Int. J. Math., Volume 25 (2014) no. 3, 1450025, 13 pages | MR 3189782 | Zbl 1295.32021

[39] Kaup, Wilhelm; Zaitsev, Dmitri On local CR-transformation of Levi-degenerate group orbits in compact Hermitian symmetric spaces, J. Eur. Math. Soc., Volume 8 (2006) no. 3, pp. 465-490 | Article | MR 2250168 | Zbl 1118.32019

[40] Khanh, Tran Vu; Pinton, Stefano; Zampieri, Giuseppe Compactness estimates for $□_{b}$ on a CR manifold, Proc. Am. Math. Soc., Volume 140 (2012) no. 9, pp. 3229-3236 | Article | Zbl 1300.32036

[41] Khanh, Tran Vu; Raich, Andrew Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type (2014) (https://arxiv.org/abs/1406.6532)

[42] Khanh, Tran Vu; Zampieri, Giuseppe Necessary geometric and analytic conditions for general estimates in the $\bar{\partial}$ -Neumann problem, Invent. Math., Volume 188 (2012) no. 3, pp. 729-750 | Article | MR 2917182 | Zbl 1362.32025

[43] Kohn, Joseph J. Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. Math., Volume 78 (1963), pp. 112-148 | Article | MR 153030 | Zbl 0161.09302

[44] Kohn, Joseph J. Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. Math., Volume 79 (1964), pp. 450-472 | Article | MR 208200 | Zbl 0178.11305

[45] Kohn, Joseph J. Boundary behavior of $\bar{\partial}$ on weakly pseudo-convex manifolds of dimension two, J. Differ. Geom., Volume 6 (1972), pp. 523-542 | Article | MR 322365 | Zbl 0256.35060

[46] Kohn, Joseph J. Subellipticity of the $\bar{\partial}$ -Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math., Volume 142 (1979) no. 1-2, pp. 79-122 | Article | MR 512213 | Zbl 0395.35069

[47] Kolář, Martin The Catlin multitype and biholomorphic equivalence of models, Int. Math. Res. Not., Volume 2010 (2010) no. 18, pp. 3530-3548 | Article | MR 2725504 | Zbl 1207.32032

[48] McNeal, Jeffery D. Lower bounds on the Bergman metric near a point of finite type, Ann. Math., Volume 136 (1992) no. 2, pp. 339-360 | Article | MR 1185122 | Zbl 0764.32006

[49] McNeal, Jeffery D. Invariant metric estimates for $\bar{\partial}$ on some pseudoconvex domains, Ark. Mat., Volume 39 (2001) no. 1, pp. 121-136 | Article | MR 1821085 | Zbl 1038.32004

[50] McNeal, Jeffery D. A sufficient condition for compactness of the $\bar{\partial}$ -Neumann operator, J. Funct. Anal., Volume 195 (2002) no. 1, pp. 190-205 | Article | MR 1934357 | Zbl 1023.32029

[51] McNeal, Jeffery D.; Varolin, Dror $L^{2}$ estimates for the $\bar{\partial}$ -operator, Bull. Math. Sci., Volume 5 (2015) no. 2, pp. 179-249 | Article | Zbl 1320.32045

[52] Nagel, Alexander; Rosay, Jean-Pierre; Stein, Elias M.; Wainger, Stephen Estimates for the Bergman and Szegö kernels in $ℂ^{2}$ , Ann. Math., Volume 129 (1989) no. 1, pp. 113-149 | Article | Zbl 0667.32016

[53] Nicoara, Andreea C. Direct proof of termination of the Kohn algorithm in the real-analytic case (2014) (https://arxiv.org/abs/1409.0963)

[54] Raich, Andrew Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann., Volume 348 (2010) no. 1, pp. 81-117 | Article | MR 2657435 | Zbl 1238.32032

[55] Raich, Andrew; Straube, Emil J. Compactness of the complex Green operator, Math. Res. Lett., Volume 15 (2008) no. 4, pp. 761-778 | Article | MR 2424911 | Zbl 1157.32032

[56] Şahutoğlu, Sönmez Strong Stein neighbourhood bases, Complex Var. Elliptic Equ., Volume 57 (2012) no. 10, pp. 1073-1085 | Article | MR 2984006 | Zbl 1267.32036

[57] Siu, Yum-Tong Multiplier ideal sheaves in complex and algebraic geometry, Sci. China, Ser. A, Volume 48 (2005), pp. 1-31 | MR 2156488 | Zbl 1131.32010

[58] Siu, Yum-Tong Effective termination of Kohn’s algorithm for subelliptic multipliers, Pure Appl. Math. Q., Volume 6 (2010) no. 4, pp. 1169-1241 | MR 2742044 | Zbl 1235.32030

[59] Siu, Yum-Tong New procedure to generate multipliers in complex Neumann problem and effective Kohn algorithm, Sci. China, Math., Volume 60 (2017) no. 6, pp. 1101-1128 | MR 3647138 | Zbl 1402.32031

[60] Straube, Emil J. Plurisubharmonic functions and subellipticity of the $\bar{\partial}$ -Neumann problem on non-smooth domains, Math. Res. Lett., Volume 4 (1997) no. 4, pp. 459-467 | Article | MR 1470417 | Zbl 0887.32005

[61] Straube, Emil J. Aspects of the $L^{2}$ -Sobolev theory of the $\bar{\partial}$ -Neumann problem, Proceedings of the international congress of mathematicians (ICM). Volume II: Invited lectures, European Mathematical Society, 2006, pp. 1453-1478 | MR 2275654 | Zbl 1106.32028

[62] Straube, Emil J. Lectures on the $L^{2}$ -Sobolev theory of the $\bar{\partial}$ -Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2010 | Zbl 1247.32003

[63] Straube, Emil J. The complex Green operator on CR-submanifolds of $ℂ^{n}$ of hypersurface type: compactness, Trans. Am. Math. Soc., Volume 364 (2012) no. 8, pp. 4107-4125 | Article | MR 2912447 | Zbl 1278.32027

[64] Webster, Sidney M. The holomorphic contact geometry of a real hypersurface, Modern methods in complex analysis (Annals of Mathematics Studies) Volume 137, Princeton University Press, 1995, pp. 327-342 | MR 1369146 | Zbl 0870.32008

[65] Yu, Ji-Ye Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J., Volume 43 (1994) no. 4, pp. 1271-1295 | MR 1322619 | Zbl 0828.32003

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT