Symmetric tensors and the geometry of subvarieties of \Bbb P^N

2006/09/18 by Fedor Bogomolov, Bruno De Oliveira, Bogomolov, Fedor +1
Engineering · Mathematics · Physics and Astronomy · #14F10 #14J10 #14N05 #32C38 #51N35 #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Complex Variables (math.CV) #Elasticity and Material Modeling #FOS: Mathematics #Tensor decomposition and applications #math.AG #math.CV #msc:14F10 #msc:14J10 #msc:14N05 #msc:32C38 #msc:51N35

paper · pdf · doi:10.48550/arxiv.math/0609471

arxiv created 2006/09/18 · openalex publication_date 2006/09/18 · arxiv updated 2011/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper following a geometric approach proves new, and reproves old, vanishing and nonvanishing results on the space of twisted symmetric differentials, H⁰(X,S^mΩ¹_X⊗ \Cal O_X(k)) with k≤ m, on subvarieties X⊂ \Bbb P^N. The case of k=m is special and the nonvanishing results are related to the space of quadrics containing X and lead to interesting geometrical objects associated to X, as for example the variety of all tangent trisecant lines of X. The same techniques give results on the symmetric differentials of subvarieties of abelian varieties. The paper ends with new results and examples about the jump along smooth families of projective varieties X_t of the symmetric plurigenera, Q_m(X)= dim H⁰(X,S^mΩ¹_X), or of the α-twisted symmetric plurigenera, Q_α,m(X)= dim H⁰(X,S^m(Ω¹_X⊗ αK_X)).

Citations

• Varieties with ample cotangent bundle

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