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Algebraic Gromov's ellipticity of cubic hypersurfaces

Shulim Kaliman, Mikhail Zaidenberg

TL;DR

The paper proves that every smooth cubic hypersurface $X\subseteq {\mathbb P}^{n+1}$ with $n\ge 2$ is elliptic in Gromov's algebraic sense, providing the first examples of irrational projective manifolds that are elliptic. The core method builds $n$ independent rank-1 sprays on $X$, obtained from lines on $X$ and a birational involution $\tau_u$, whose orbit directions span the tangent space at a general point, thereby establishing local ellipticity and, by Gr's Localization Lemma, global ellipticity; an alternative proof uses Kusakabe's surjectivity to construct a dominating spray system. As a corollary, the punctured affine cone over $X$ is elliptic, and there are surjective morphisms from affine spaces to $X$ with properties compatible with ellipticity. Overall, the work extends the class of known elliptic varieties beyond uniformly rational examples and informs the relationship between ellipticity and rationality in projective geometry.

Abstract

We show that every smooth cubic hypersurface X in P^{n+1}, n> 1 is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured affine cone over X is elliptic.

Algebraic Gromov's ellipticity of cubic hypersurfaces

TL;DR

The paper proves that every smooth cubic hypersurface with is elliptic in Gromov's algebraic sense, providing the first examples of irrational projective manifolds that are elliptic. The core method builds independent rank-1 sprays on , obtained from lines on and a birational involution , whose orbit directions span the tangent space at a general point, thereby establishing local ellipticity and, by Gr's Localization Lemma, global ellipticity; an alternative proof uses Kusakabe's surjectivity to construct a dominating spray system. As a corollary, the punctured affine cone over is elliptic, and there are surjective morphisms from affine spaces to with properties compatible with ellipticity. Overall, the work extends the class of known elliptic varieties beyond uniformly rational examples and informs the relationship between ellipticity and rationality in projective geometry.

Abstract

We show that every smooth cubic hypersurface X in P^{n+1}, n> 1 is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured affine cone over X is elliptic.
Paper Structure (4 sections, 15 theorems, 10 equations)

This paper contains 4 sections, 15 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Criteria of ellipticity
  3. Ellipticity of cubic hypersurfaces
  4. An alternative proof

Key Result

Theorem 1.1

A smooth cubic hypersurface $X\subseteq{\mathbb P}^{n+1}$ of dimension $n\ge 2$ is elliptic in the sense of Gromov.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.4
  • Proposition 2.1: Gromov's extension lemma
  • Corollary 2.2: Gromov's Localization Lemma
  • proof : Sketch of the proof
  • Remark 2.3
  • Proposition 2.4: KZ1
  • Lemma 2.5
  • proof : Sketch of the proof
  • Lemma 3.2
  • ...and 18 more