Mathematische Zeitschrift

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

Initial monomial invariants of holomorphic maps

  • Dusty Grundmeier
  • Jiří Lebl


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.


Vector Subspace Monomial Order Connected Complex Manifold Generic Initial Ideal Affine Span 
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© 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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