Open Access
January, 2021 Newton polyhedra and order of contact on real hypersurfaces
Joe KAMIMOTO
J. Math. Soc. Japan 73(1): 1-39 (January, 2021). DOI: 10.2969/jmsj/80868086
Abstract

The purpose of this paper is to investigate order of contact on real hypersurfaces in $\mathbb{C}^n$ by using Newton polyhedra which are important notion in the study of singularity theory. To be more precise, an equivalence condition for the equality of regular type and singular type is given by using the Newton polyhedron of a defining function for the respective hypersurface. Furthermore, a sufficient condition for this condition, which is more useful, is also given. This sufficient condition is satisfied by many earlier known cases (convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4, etc.). Under the above conditions, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

References

1.

[1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps I, Monogr. Math., 82, Birkhäuser, 1985. 1290.58001[1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps I, Monogr. Math., 82, Birkhäuser, 1985. 1290.58001

2.

[2] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps II, Monogr. Math., 83, Birkhäuser, 1988. 1290.58001[2] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps II, Monogr. Math., 83, Birkhäuser, 1988. 1290.58001

3.

[3] G. Bharali and B. Stensønes, Plurisubharmonic polynomials and bumping, Math. Z., 261 (2009), 39–63.[3] G. Bharali and B. Stensønes, Plurisubharmonic polynomials and bumping, Math. Z., 261 (2009), 39–63.

4.

[4] H. P. Boas and E. J. Straube, On equality of line type and variety type of real hypersurfaces in $\mathbb{C}^n$, J. Geom. Anal., 2 (1992), 95–98. 0749.32009 10.1007/BF02921382[4] H. P. Boas and E. J. Straube, On equality of line type and variety type of real hypersurfaces in $\mathbb{C}^n$, J. Geom. Anal., 2 (1992), 95–98. 0749.32009 10.1007/BF02921382

5.

[5] D. Catlin, Necessary conditions for subellipticity of the $\bar{\partial}$-Neumann problem, Ann. of Math. (2), 117 (1983), 147–171. 0552.32017 10.2307/2006974[5] D. Catlin, Necessary conditions for subellipticity of the $\bar{\partial}$-Neumann problem, Ann. of Math. (2), 117 (1983), 147–171. 0552.32017 10.2307/2006974

6.

[6] D. Catlin, Subelliptic estimates for the $\bar{\partial}$-Neumann problem on pseudoconvex domains, Ann. of Math. (2), 126 (1987), 131–191. 0627.32013 10.2307/1971347[6] D. Catlin, Subelliptic estimates for the $\bar{\partial}$-Neumann problem on pseudoconvex domains, Ann. of Math. (2), 126 (1987), 131–191. 0627.32013 10.2307/1971347

7.

[7] J. P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2), 115 (1982), 615–637. 0488.32008 10.2307/2007015[7] J. P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2), 115 (1982), 615–637. 0488.32008 10.2307/2007015

8.

[8] J. P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, Stud. Adv. Math., CRC Press, 1993. MR1224231[8] J. P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, Stud. Adv. Math., CRC Press, 1993. MR1224231

9.

[9] J. P. D'Angelo, A remark on finite type conditions, J. Geom. Anal., 28 (2018), 2602–2608. 1423.32017 10.1007/s12220-017-9921-1[9] J. P. D'Angelo, A remark on finite type conditions, J. Geom. Anal., 28 (2018), 2602–2608. 1423.32017 10.1007/s12220-017-9921-1

10.

[10] B.-Y. Chen and S. Fu, The reproducing kernels and the finite type conditions, Illinois J. Math., 56 (2012), 67–83. 1278.32005 10.1215/ijm/1380287460 euclid.ijm/1380287460[10] B.-Y. Chen and S. Fu, The reproducing kernels and the finite type conditions, Illinois J. Math., 56 (2012), 67–83. 1278.32005 10.1215/ijm/1380287460 euclid.ijm/1380287460

11.

[11] B.-Y. Chen, J. Kamimoto and T. Ohsawa, Behavior of the Bergman kernel at infinity, Math. Z., 248 (2004), 695–708. 1066.32003 10.1007/s00209-004-0676-6[11] B.-Y. Chen, J. Kamimoto and T. Ohsawa, Behavior of the Bergman kernel at infinity, Math. Z., 248 (2004), 695–708. 1066.32003 10.1007/s00209-004-0676-6

12.

[12] K. Diederich and G. Herbort, Pseudoconvex domains of semiregular type, In: Contributions to Complex Analysis and Analytic Geometry, Aspects Math., E26, Friedr. Vieweg, Braunschweig, 1994, 127–161. 0845.32019[12] K. Diederich and G. Herbort, Pseudoconvex domains of semiregular type, In: Contributions to Complex Analysis and Analytic Geometry, Aspects Math., E26, Friedr. Vieweg, Braunschweig, 1994, 127–161. 0845.32019

13.

[13] J. E. Fornæss and B. Stensønes, Maximally tangent complex curves for germs of finite type $\mathcal{C}^{\infty}$ pseudoconvex domains in $\mathbb{C}^3$, Math. Ann., 347 (2010), 979–991.[13] J. E. Fornæss and B. Stensønes, Maximally tangent complex curves for germs of finite type $\mathcal{C}^{\infty}$ pseudoconvex domains in $\mathbb{C}^3$, Math. Ann., 347 (2010), 979–991.

14.

[14] S. Fu, A. V. Isaev and S. G. Krantz, Finite type conditions on Reinhardt domains, Complex Variables Theory Appl., 31 (1996), 357–363. 0955.32028 10.1080/17476939608814973[14] S. Fu, A. V. Isaev and S. G. Krantz, Finite type conditions on Reinhardt domains, Complex Variables Theory Appl., 31 (1996), 357–363. 0955.32028 10.1080/17476939608814973

15.

[15] T. Fukui, Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc., 112 (1991), 1169–1183. 0737.58001[15] T. Fukui, Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc., 112 (1991), 1169–1183. 0737.58001

16.

[16] R. C. Gunning, Lectures on Complex Analytic Varieties: The Local Parametrization Theorem, Math. Notes Princeton Univ. Press, Princeton, NJ, 1970. 0213.35904[16] R. C. Gunning, Lectures on Complex Analytic Varieties: The Local Parametrization Theorem, Math. Notes Princeton Univ. Press, Princeton, NJ, 1970. 0213.35904

17.

[17] G. Heier, Finite type and the effective Nullstellensatz, Comm. Algebra, 36 (2008), 2947–2957. 1149.14047 10.1080/00927870802110342[17] G. Heier, Finite type and the effective Nullstellensatz, Comm. Algebra, 36 (2008), 2947–2957. 1149.14047 10.1080/00927870802110342

18.

[18] I. A. Ikromov and D. Müller, Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra, Ann. of Math. Stud., 194, Princeton Univ. Press, Princeton, NJ, 2016.[18] I. A. Ikromov and D. Müller, Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra, Ann. of Math. Stud., 194, Princeton Univ. Press, Princeton, NJ, 2016.

19.

[19] Y. Jikumaru, Master thesis (Japanese), Kyushu Univ., 2017.[19] Y. Jikumaru, Master thesis (Japanese), Kyushu Univ., 2017.

20.

[20] J. Kamimoto, Newton polyhedra and the Bergman kernel, Math. Z., 246 (2004), 405–440. 1047.32004 10.1007/s00209-003-0554-7[20] J. Kamimoto, Newton polyhedra and the Bergman kernel, Math. Z., 246 (2004), 405–440. 1047.32004 10.1007/s00209-003-0554-7

21.

[21] J. Kamimoto, A sufficient condition for equality of regular type and singular type of real hypersurfaces, to appear in Sūrikaisekikenkyūsho Kōkyūroku.[21] J. Kamimoto, A sufficient condition for equality of regular type and singular type of real hypersurfaces, to appear in Sūrikaisekikenkyūsho Kōkyūroku.

22.

[22] J. Kamimoto and T. Nose, Newton polyhedra and weighted oscillatory integrals with smooth phases, Trans. Amer. Math. Soc., 368 (2016), 5301–5361. 1335.58025 10.1090/tran/6528[22] J. Kamimoto and T. Nose, Newton polyhedra and weighted oscillatory integrals with smooth phases, Trans. Amer. Math. Soc., 368 (2016), 5301–5361. 1335.58025 10.1090/tran/6528

23.

[23] J. J. Kohn, Subellipticity of the $\bar{\partial}$-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math., 142 (1979), 79–122. 0395.35069 10.1007/BF02395058 euclid.acta/1485890016[23] J. J. Kohn, Subellipticity of the $\bar{\partial}$-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math., 142 (1979), 79–122. 0395.35069 10.1007/BF02395058 euclid.acta/1485890016

24.

[24] J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265–268. 0248.32013 10.1007/BF01428194[24] J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265–268. 0248.32013 10.1007/BF01428194

25.

[25] M. Kolář, Convexifiability and supporting functions in $\mathbb{C}^2$, Math. Res. Lett., 2 (1995), 505–513. 0846.32011 10.4310/MRL.1995.v2.n4.a10[25] M. Kolář, Convexifiability and supporting functions in $\mathbb{C}^2$, Math. Res. Lett., 2 (1995), 505–513. 0846.32011 10.4310/MRL.1995.v2.n4.a10

26.

[26] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1–31.[26] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1–31.

27.

[27] B. Lichtin, Estimation of Łojasiewicz exponents and Newton polygons, Invent. Math., 64 (1981), 417–429. 0556.32003 10.1007/BF01389274[27] B. Lichtin, Estimation of Łojasiewicz exponents and Newton polygons, Invent. Math., 64 (1981), 417–429. 0556.32003 10.1007/BF01389274

28.

[28] M. Lejeune-Jalabert and B. Teissier, Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 781–859.[28] M. Lejeune-Jalabert and B. Teissier, Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 781–859.

29.

[29] J. D. McNeal, Convex domains of finite type, J. Funct. Anal., 108 (1992), 361–373. 0777.31007 10.1016/0022-1236(92)90029-I[29] J. D. McNeal, Convex domains of finite type, J. Funct. Anal., 108 (1992), 361–373. 0777.31007 10.1016/0022-1236(92)90029-I

30.

[30] J. D. McNeal and L. Mernik, Regular versus singular order of contact on pseudoconvex hypersurfaces, J. Geom. Anal., 28 (2018), 2653–2669. 1417.32039 10.1007/s12220-017-9926-9[30] J. D. McNeal and L. Mernik, Regular versus singular order of contact on pseudoconvex hypersurfaces, J. Geom. Anal., 28 (2018), 2653–2669. 1417.32039 10.1007/s12220-017-9926-9

31.

[31] J. D. McNeal and A. Némethi, The order of contact of a holomorphic ideal in $\mathbb{C}^2$, Math. Z., 250 (2005), 873–883.[31] J. D. McNeal and A. Némethi, The order of contact of a holomorphic ideal in $\mathbb{C}^2$, Math. Z., 250 (2005), 873–883.

32.

[32] M. Oka, Non-Degenerate Complete Intersection Singularity, Actualités Math., Hermann, Paris, 1997.[32] M. Oka, Non-Degenerate Complete Intersection Singularity, Actualités Math., Hermann, Paris, 1997.

33.

[33] G. Oleksik, The Łojasiewicz exponent of nondegenerate surface singularities, Acta Math. Hungar., 138 (2013), 179–199. 1289.32022 10.1007/s10474-012-0285-5[33] G. Oleksik, The Łojasiewicz exponent of nondegenerate surface singularities, Acta Math. Hungar., 138 (2013), 179–199. 1289.32022 10.1007/s10474-012-0285-5

34.

[34] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math., 179 (1997), 105–152. 0896.35147 10.1007/BF02392721 euclid.acta/1485891072[34] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math., 179 (1997), 105–152. 0896.35147 10.1007/BF02392721 euclid.acta/1485891072

35.

[35] N. Uehara, Master thesis (Japanese), Kyushu Univ., 2013.[35] N. Uehara, Master thesis (Japanese), Kyushu Univ., 2013.

36.

[36] J. Y. Yu, Multitypes of convex domains, Indiana Univ. Math. J., 41 (1992), 837–849. 0759.32009 10.1512/iumj.1992.41.41044[36] J. Y. Yu, Multitypes of convex domains, Indiana Univ. Math. J., 41 (1992), 837–849. 0759.32009 10.1512/iumj.1992.41.41044

37.

[37] J. Y. Yu, Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J., 43 (1994), 1271–1295. 0828.32003 10.1512/iumj.1994.43.43055[37] J. Y. Yu, Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J., 43 (1994), 1271–1295. 0828.32003 10.1512/iumj.1994.43.43055

38.

[38] A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Functional Anal. Appl., 10 (1976), 175–196. 0351.32011 10.1007/BF01075524[38] A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Functional Anal. Appl., 10 (1976), 175–196. 0351.32011 10.1007/BF01075524

39.

[39] D. Zaitsev, A geometric approach to Catlin's boundary systems, Ann. Inst. Fourier (Grenoble), 69 (2019), 2635–2679. 1436.32101 10.5802/aif.3304[39] D. Zaitsev, A geometric approach to Catlin's boundary systems, Ann. Inst. Fourier (Grenoble), 69 (2019), 2635–2679. 1436.32101 10.5802/aif.3304

40.

[40] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995.[40] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995.
Copyright © 2021 Mathematical Society of Japan
Joe KAMIMOTO "Newton polyhedra and order of contact on real hypersurfaces," Journal of the Mathematical Society of Japan 73(1), 1-39, (January, 2021). https://doi.org/10.2969/jmsj/80868086
Received: 8 July 2018; Published: January, 2021
Vol.73 • No. 1 • January, 2021
Back to Top