On non-proper intersections and local intersection numbers

2020/03/13 by Andersson, Mats, Kalm, Håkan Samuelsson, Wulcan, Elizabeth · 1 citation
#14C17 #32Q99 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.06180

Abstract

Given pure-dimensional (generalized) cycles μ₁ and μ₂ on a complex manifold Y we introduce a product μ₁\diamond_Y μ₂ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. % If Y is projective, then given a very ample line bundle L→ Y we define a product μ₁\bl μ₂ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that μ₁ and μ₂ are effective, this product satisfies a Bézout inequality. If i\colon Y→ \Pk^N is an embedding such that i^*\Ok(1)=L, then μ₁\bl μ₂ can be expressed as a mean value of Stückrad-Vogel cycles on \Pk^N. There are quite explicit relations between \di_Y and \bl.

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