Regularity on abelian varieties I.

*(English)*Zbl 1022.14012The 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.

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.

Reviewer: Georg Hein (Berlin)

##### MSC:

14K12 | Subvarieties of abelian varieties |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14H40 | Jacobians, Prym varieties |

14K05 | Algebraic theory of abelian varieties |

##### Keywords:

globally generated tensor products of sheaves; abelian variety; Mukai regularity; Fourier-Mukai transform
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\textit{G. Pareschi} and \textit{M. Popa}, J. Am. Math. Soc. 16, No. 2, 285--302 (2003; Zbl 1022.14012)

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