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), either the image of the mapping is contained in a complex affine subspace, or A is bounded by a constant depending only on B. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both A and B are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.

Initial monomial invariants of holomorphic maps

“…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).…”

“…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 , .…”

“…In this section we briefly introduce the relevant definitions and results from commutative algebra. 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.…”

Invariant CR mappings between hyperquadrics

“…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.…”

An application of Macaulay’s estimate to sums of squares problems in several complex variables

“…, 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.…”