2013
DOI: 10.1007/s00208-013-0989-z
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Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Abstract: Abstract. Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi-Huang and Baouendi-Ebenfelt-Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric Q (A, B)… Show more

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Cited by 15 publications
(11 citation statements)
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“…To generalize these results to arbitrary holomorphic maps we improve a result proved in Grundmeier-Lebl-Vivas [13] that itself is a version of Galligo's theorem for vector subspaces of O(U). Given a subspace X ⊂ O(U), we precompose with a generic affine map τ , and define the generic initial monomial subspace as the space spanned by the initial monomials of elements of X • τ , denoted gin(X).…”
Section: Introductionmentioning
confidence: 92%
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“…To generalize these results to arbitrary holomorphic maps we improve a result proved in Grundmeier-Lebl-Vivas [13] that itself is a version of Galligo's theorem for vector subspaces of O(U). Given a subspace X ⊂ O(U), we precompose with a generic affine map τ , and define the generic initial monomial subspace as the space spanned by the initial monomials of elements of X • τ , denoted gin(X).…”
Section: Introductionmentioning
confidence: 92%
“…We use the setup from Green [12], and in a later section we use the techniques developed by the authors in [13]. To be consistent with these two papers, we use the slightly unusual monomial ordering as used by Green. Let Z 0 , Z 1 , .…”
Section: Generic Initial Ideals and Rational Mapsmentioning
confidence: 99%
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“…The problem of constructing group-invariant CR mappings has attracted substantial interest over the years (see [2][3][4][6][7][8][9][10][11][12][13] and the references therein). D'Angelo and Lichtblau [5] gave a canonical construction of invariant-polynomial CR mappings, and they used this construction to study the CR spherical space form problem.…”
Section: Introductionmentioning
confidence: 99%
“…, z n ], and we use a well-known estimate of Macaulay (Theorem 1 below). Ours is not the first paper to apply such results from commutative algebra to questions in CR geometry; Grundmeier, Lebl, and Vivas use a similar set of ideas in [9] to prove a rigidity theorem for CR mappings of hyperquadrics. Our goal is not primarily to obtain specific inequalities but rather to illustrate how a set of ideas from this area of algebra can be brought to bear on questions arising in CR geometry.…”
Section: Introductionmentioning
confidence: 99%