Bézout's theorem for abelian varieties

2025/09/18 by Debarre, Olivier, Moonen, Ben
#14F20) #14K05 (14F06 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.14940

Abstract

Let X, Y be closed irreducible subvarieties of an absolutely simple abelian variety of dimension g over a field. If dim(X) + dim(Y) ≤ g, we prove that the addition morphism X × Y → X + Y is semismall. As a consequence, we deduce that if dim(X) + dim(Y) ≥ g, the subvarieties X and Y must meet (Bézout's theorem). If we drop the assumption that the abelian variety is absolutely simple, we prove that Bézout's theorem still holds if X satisfies a nondegeneracy condition. These results were previously known only in characteristic zero. Our proof of the semismallness statement is based on the theory of perverse sheaves: using results of Krämer and Weissauer, we prove that for perverse sheaves K supported on X, and L supported on Y, the convolution product K * L is again perverse.

Citations

• Constructible sheaves on schemes
• The monodromy of families of subvarieties on abelian varieties
• Relative Perversity
• The Shafarevich conjecture for hypersurfaces in abelian varieties
• The characteristic cycle and the singular support of a constructible sheaf
• The pro-étale topology for schemes
• Why certain Tannaka groups attached to abelian varieties are almost\n connected
• The Decomposition Theorem and the topology of algebraic maps
• On a conjecture of Kashiwara
• Subvarieties of abelian varieties
• Geometric Eisenstein series
• The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties
• Theoremes de connexites et varietes abeliennes

Related

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• Perverse sheaves on semi-abelian varieties -- a survey of properties and applications
• Reconstructing Abelian Varieties via Model Theory
• Minimal abelian varieties of algebras, I
• Weak semistable reduction in characteristic 0