Regularity on abelian varieties I

By Giuseppe Pareschi, Mihnea Cristian Popa. 2001

154 citations recorded by the source corpus.

We introduce the notion of Mukai regularity ($M$-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/63e6234d015e58b99b7c7d95b70a4b096e20a8a7

How researchers see this paper

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In Computational Theory and Mathematics

1. Gregory G. Smith
2. Diane Maclagan
3. Mihnea Cristian Popa
4. Giuseppe Pareschi
5. P. Stellari
6. Arend Bayer
7. Emanuele Macrì
8. Ali Soleyman Jahan

In Geometry and Topology

1. Emanuele Macrì
2. Arend Bayer
3. P. Stellari
4. Mihnea Cristian Popa
5. Gregory G. Smith
6. Chunyi Li
7. Giuseppe Pareschi
8. Diane Maclagan

In Statistical and Nonlinear Physics

1. Gregory G. Smith
2. Mihnea Cristian Popa
3. Giuseppe Pareschi
4. P. Stellari
5. Arend Bayer
6. Diane Maclagan
7. Emanuele Macrì

Citations in the corpus

• Syzygies, multigraded regularity and toric varieties (2005, background): “Explicit conditions certifying that an ample line bundle is suitably positive are given by Green [9] for curves, Ein and Lazarsfeld [3] for smooth varieties, Gallego and Purnaprajna [8] for normal surfaces, and Pareschi and Popa [25, 27, 26] for abelian varieties; see Lazarsfeld [16, §1.”
• GV-sheaves, Fourier-Mukai transform, and generic vanishing (2006, background): “In [ PP1 ] Proposition 4.4, we proved that the sheavesOWd() are M-regular and h 0(OWd() ) = 1 for 2 Pic0(X)”
• Multigraded Castelnuovo-Mumford Regularity (2003, background): “On the other hand, the versions of regularity developed for Grassmannians in [9] and abelian varieties in [33] do not fall under the umbrella of multigraded regularity.”
• The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, background): “For abelian varieties, much stronger statements are known, see [37,38].”