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The algebra of symmetric tensors on smooth projective varieties

Arnaud Beauville, Jie Liu

Abstract

We discuss in this note the algebra H^0(X, Sym*TX) for a smooth complex projective variety X . We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.

The algebra of symmetric tensors on smooth projective varieties

Abstract

We discuss in this note the algebra H^0(X, Sym*TX) for a smooth complex projective variety X . We compute it in some simple examples, and give a sharp bound on its Krull dimension. Then we propose a conjectural characterization of non-uniruled projective manifolds with pseudo-effective tangent bundle.
Paper Structure (19 sections, 17 theorems, 18 equations)

This paper contains 19 sections, 17 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Some examples
  3. Abelian varieties
  4. Projective space
  5. Rational homogeneous manifolds
  6. Flag varieties, Grassmannians
  7. Quadrics
  8. Intersection of two quadrics
  9. Completely integrable systems
  10. An example: ruled surfaces
  11. Cases with $S(X)=\mathbb{C}$
  12. Varieties with $c_1(X)=0$
  13. Varieties of general type
  14. Hypersurfaces
  15. The Krull dimension of S(X)
  16. ...and 4 more sections

Key Result

Proposition 1

The graded algebra $S(\mathbb{P}(V))$ is isomorphic to the quotient of $\bigoplus_{d\geq 0}\bigl(\mathsf{S}^dV\otimes \mathsf{S}^dV^*\bigr)$ by the ideal generated by $I$.

Theorems & Definitions (25)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 15 more