Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

Grieve, Nathan

doi:10.2478/aupcsm-2021-0007

Abstract

We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endo-morphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

References

[1] Atiyah, Michael F. “Vector bundles over an elliptic curve.” Proc. London Math. Soc. (3) 7 (1957): 414-452. Cited on 105.10.1112/plms/s3-7.1.414
Search in Google Scholar Back to article
[2] Barja, Miguel Ángel and Rita Pardini, and Lidia Stoppino. “Linear systems on irregular varieties.” J. Inst. Math. Jussieu 19, no. 6 (2020): 2087-2125. Cited on 96.10.1017/S1474748019000069
Search in Google Scholar Back to article
[3] Bayer, David and Michael Stillman. “A criterion for detecting m-regularity.” Invent. Math. 87, no. 1 (1987): 1-11. Cited on 96.10.1007/BF01389151
Search in Google Scholar Back to article
[4] Brion, Michel. “Homogeneous projective bundles over abelian varieties.” Algebra Number Theory 7, no. 10 (2013): 2475-2510. Cited on 106.10.2140/ant.2013.7.2475
Search in Google Scholar Back to article
[5] Green, Mark and Robert Lazarsfeld. “Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville.” Invent. Math. 90, no. 2 (1987): 389-407. Cited on 96.10.1007/BF01388711
Search in Google Scholar Back to article
[6] Grieve, Nathan. “Index conditions and cup-product maps on Abelian varieties.” Internat. J. Math. 25, no. 4 (2014): 1450036, 31. Cited on 97, 98, 100, 107 and 113.10.1142/S0129167X14500360
Search in Google Scholar Back to article
[7] Grieve, Nathan. “Refinements to Mumford’s theta and adelic theta groups.” Ann. Math. Qué. 38, no. 2 (2014): 145-167. Cited on 97 and 106.10.1007/s40316-014-0024-0
Search in Google Scholar Back to article
[8] Grieve, Nathan. “Reduced norms and the Riemann-Roch theorem for Abelian varieties.” New York J. Math. 23 (2017): 1087–1110. Cited on 97, 98, 99, 101, 103, 104, 105, 110 and 115.
Search in Google Scholar Back to article
[9] Grieve, Nathan. “On cubic torsors, biextensions and Severi-Brauer varieties over Abelian varieties.” São Paulo J. Math. Sci. (to appear). Cited on 106.
Search in Google Scholar Back to article
[10] Gulbrandsen, Martin G. “Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties.” Matematiche (Catania) 63, no. 1 (2008): 123-137. Cited on 101, 107, 109, 114 and 115.
Search in Google Scholar Back to article
[11] Hacon, Christopher D. “A derived category approach to generic vanishing.” J. Reine Angew. Math. 575 (2004): 173-187. Cited on 96.10.1515/crll.2004.078
Search in Google Scholar Back to article
[12] Hulek, Klaus and Roberto Laface. “On the Picard numbers of Abelian varieties.” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19, no. 3 (2019): 1199-1224. Cited on 102.10.2422/2036-2145.201706_007
Search in Google Scholar Back to article
[13] Küronya, Alex and Yusuf Mustopa. “Continuous CM-regularity of semihomogeneous vector bundles”. Adv. Geom. 20, no. 3 (2020): 401-412. Cited on 96, 97, 100, 109, 111, 113, 114 and 116.10.1515/advgeom-2019-0011
Search in Google Scholar Back to article
[14] Mukai, Shigeru. “Semi-homogeneous vector bundles on an Abelian variety.” J. Math. Kyoto Univ. 18, no. 2 (1978): 239-272. Cited on 96, 105, 106, 114 and 116.10.1215/kjm/1250522574
Search in Google Scholar Back to article
[15] Mukai, Shigeru. “Duality between D(X) and D(X̂) with its application to Picard sheaves.” Nagoya Math. J. 81 (1981): 153-175. Cited on 96.10.1017/S002776300001922X
Search in Google Scholar Back to article
[16] Mumford, David. Lectures on curves on an algebraic surface. Vol. 59 of Annals of Mathematics Studies. Princeton, N.J.: Princeton University Press, 1966. Cited on 96.10.1515/9781400882069
Search in Google Scholar Back to article
[17] Mumford, David. Abelian varieties. Vol. 5 of Tata Institute of Fundamental Research Studies in Mathematics. Bombay; Oxford University Press, London: Published for the Tata Institute of Fundamental Research, 1970. Cited on 97, 98, 101, 103, 105 and 111.
Search in Google Scholar Back to article
[18] Mumford, David. “Varieties defined by quadratic equations.” Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), 29-100. Rome: Edizioni Cremonese, 1970. Cited on 97, 98, 101, 107, 108, 111, 113 and 115.
Search in Google Scholar Back to article
[19] Lazarsfeld, Robert. Positivity in algebraic geometry. I, Vol. 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer-Verlag, 2004. Cited on 96.
Search in Google Scholar Back to article
[20] Pareschi, Giuseppe and Mihnea Popa. “Regularity on abelian varieties. I.” J. Amer. Math. Soc. 16, no. 2 (2003): 285-302. Cited on 96, 110 and 111.10.1090/S0894-0347-02-00414-9
Search in Google Scholar Back to article
[21] Pareschi, Giuseppe and Mihnea Popa. “Regularity on abelian varieties. II. Basic results on linear series and defining equations.” J. Algebraic Geom. 13, no. 1 (2004): 167-193. Cited on 96.10.1090/S1056-3911-03-00345-X
Search in Google Scholar Back to article
[22] Pareschi, Giuseppe and Mihnea Popa. “Regularity on abelian varieties III: relationship with generic vanishing and applications.” Grassmannians, moduli spaces and vector bundles, 141-167. Providence, RI, Amer. Math. Soc., 2011. Cited on 96.
Search in Google Scholar Back to article

Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

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