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Regularity on abelian varieties I. (English) Zbl 1022.14012
The authors give a criterion for tensor products of sheaves on a abelian variety to be globally generated in terms of the Mukai regularity (M-regularity). In addition they give several applications of their main result. They use the following notations: $$X$$ an abelian variety of dimension $$g$$ over an algebraically closed field, $$\hat X$$ its dual, {$$\mathcal P$$} a Poincaré bundle on $$X \times \hat X$$, $$p_X$$ and $$p_{\hat X}$$ the two projections from $$X \times \hat X$$. A coherent sheaf $$F$$ on $$X$$ is called M-regular, if the codimension of the support of $$R^ip_{\hat X*}({\mathcal P} \otimes p_X^*F)$$ is greater than $$i$$ for all $$i=1,\ldots g$$.
The main result (theorem 2.4) of the article is the M-regularity criterion which says, that the tensor product $$F \otimes \iota_*L$$ of two M-regular sheaves $$F$$ and $$\iota_*L$$ is globally generated for $$L$$ an invertible sheaf on a closed subvariety $$\iota:Y \to X$$.
The proof of this theorem uses the Fourier-Mukai transform between the derived categories $$D(X)$$ and $$D(\hat X)$$ and the concept of continuous global generation. The applications are results on base point freeness, very ampleness, vanishing results, and dimensions of spaces of global sections.

MSC:
 14K12 Subvarieties of abelian varieties 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14H40 Jacobians, Prym varieties 14K05 Algebraic theory of abelian varieties
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References:
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