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Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 371–387 | Cite as

Initial monomial invariants of holomorphic maps

  • Dusty Grundmeier
  • Jiří Lebl
Article
  • 95 Downloads

Abstract

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics, giving a readily computable invariant in this important case. For example, the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball. Next, we associate to each subspace \(X \subset {\mathcal {O}}(U)\) a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics.

Keywords

Vector Subspace Monomial Order Connected Complex Manifold Generic Initial Ideal Affine Span 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alexander, H.: Proper holomorphic mappings in \(C^{n}\). Indiana Univ. Math. J. 26(1), 137–146 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)zbMATHGoogle Scholar
  3. 3.
    D’Angelo, J.P.: Polynomial proper maps between balls. Duke Math. J. 57(1), 211–219 (1988). doi: 10.1215/S0012-7094-88-05710-9 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Angelo, J.P.: Hermitian Analysis, Cornerstones. From Fourier Series to Cauchy-Riemann Geometry. Birkhäuser, New York (2013)zbMATHGoogle Scholar
  5. 5.
    D’Angelo, J.P., Kos, Š., Riehl, E.: A sharp bound for the degree of proper monomial mappings between balls. J. Geom. Anal. 13(4), 581–593 (2003). doi: 10.1007/BF02921879 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Faran, J.J.: Maps from the two-ball to the three-ball. Invent. Math. 68(3), 441–475 (1982). doi: 10.1007/BF01389412 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Faran, J., Huang, X., Ji, S., Zhang, Y.: Polynomial and rational maps between balls. Pure Appl. Math. Q. 6(3), 829–842 (2010). doi: 10.4310/PAMQ.2010.v6.n3.a10 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Forstnerič, F.: Embedding strictly pseudoconvex domains into balls. Trans. Am. Math. Soc. 295(1), 347–368 (1986). doi: 10.2307/2000160 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Forstnerič, F.: Extending proper holomorphic mappings of positive codimension. Invent. Math. 95(1), 31–61 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grauert, H.: Über die Deformation isolierter Singularitäten analytischer Mengen. Invent. Math. 15, 171–198 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grayson, D. R., Stillman, M. E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  12. 12.
    Green, M.: Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166. Birkhäuser, Basel, 1998, pp. 119–186Google Scholar
  13. 13.
    Grundmeier, D., Lebl, J., Vivas, L.: Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics. Math. Ann. 358(3–4), 1059–1089 (2014). doi: 10.1007/s00208-013-0989-z MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huang, X., Ji, S., Yin, W.: On the third gap for proper holomorphic maps between balls. Math. Ann. 358(1–2), 115–142 (2014). doi: 10.1007/s00208-013-0952-z MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lebl, J.: Hermitian operators, and CR maps of spheres and hyperquadrics. Michigan Math. J. 60(3), 603–628 (2011). doi: 10.1307/mmj/1320763051 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reiter, M.: Classification of holomorphic mappings of hyperquadrics from \(\mathbb{C}^2\). J. Geom. Anal., arXiv:1409.5968
  17. 17.
    Webster, S.M.: Some birational invariants for algebraic real hypersurfaces. Duke Math. J. 45(1), 39–46 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsBall State UniversityMuncieUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Department of MathematicsOklahoma State UniversityStillwaterUSA

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