Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T15:28:14.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

Published online by Cambridge University Press:  27 February 2013

ANA CRISTINA LÓPEZ MARTÍN ,
DARÍO SÁNCHEZ GÓMEZ  and
CARLOS TEJERO PRIETO
ANA CRISTINA LÓPEZ MARTÍN
Affiliation:
Departamento de Matemáticas and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM), Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain. e-mail: [email protected], [email protected], [email protected]
DARÍO SÁNCHEZ GÓMEZ
Affiliation:
Departamento de Matemáticas and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM), Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain. e-mail: [email protected], [email protected], [email protected]
CARLOS TEJERO PRIETO
Affiliation:
Departamento de Matemáticas and Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM), Universidad de Salamanca, Plaza de la Merced 1–4, 37008 Salamanca, Spain. e-mail: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the group of relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier–Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier–Mukai transforms and we prove that the number of relative Fourier–Mukai partners of a given abelian scheme over a normal base is finite.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 129 - 153
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andreas, B. and Ruipérez, D. HernándezComments on N=1 heterotic string vacua. Adv. Theor. Math. Phys. 7 (2003), pp. 751–786.CrossRefGoogle Scholar
[2]Andreas, B., Ruipérez, D. Hernández and Gómez, D. SánchezStable sheaves over K3 fibrations. Internat. J. Math. 21 (2010), pp. 25–46.CrossRefGoogle Scholar
[3]Andreas, B., Yau, S.-T., Curio, G. and Ruipérez, D. H. Fibrewise T-duality for D-branes on elliptic Calabi-Yau. J. High Energy Phys. (2001), pp. Paper 20, 13.Google Scholar
[4]Ballard, R. M. Equivalences of derived categories of sheaves on quasi-projective schemes. math. AG. 09053148v2.Google Scholar
[5]Ballard, R. M.Derived categories of sheaves on singular schemes with an application to reconstruction. Adv. Math. 227 (2011), pp. 895–919.CrossRefGoogle Scholar
[6]Bartocci, C., Bruzzo, U. and Ruipérez, D. HernándezFourier-Mukai and Nahm transforms in geometry and mathematical physics. Progr. Math. vol. 276 (Birkhäuser Boston Inc. (Boston, MA, 2009).Google Scholar
[7]Behrens, M. and Lawson, T.Topological automorphic forms. Mem. Amer. Math. Soc. 204 (2010), pp. xxiv+141.Google Scholar
[8]Bondal, A. and Orlov, D.Derived categories of coherent sheaves. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Higher Ed. Press), pp. 47–56.Google Scholar
[9]Bondal, A. I. and Orlov, D. O.Reconstruction of a variety from the derived category and groups of autoequivalences. Composit. Math. 125 (2001), pp. 327–344.CrossRefGoogle Scholar
[10]Bridgeland, T.Equivalences of triangulated categories and Fourier-Mukai transforms. Bull. London Math. Soc. 31 (1999), pp. 25–34.CrossRefGoogle Scholar
[11]Broomhead, N. and Ploog, D. Autoequivalences of toric surfaces. math.AG. 1010.1717v1.Google Scholar
[12]Burban, I. and Kreussler, B.Derived categories of irreducible projective curves of arithmetic genus one. Composit. Math. 142 (2006), pp. 1231–1262.CrossRefGoogle Scholar
[13]Canonaco, A. and Stellari, P. Fourier–Mukai functors: a survey. math. AG. 1109.3083.Google Scholar
[14]Deligne, P. and Pappas, G.Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compositi. Math. 90 (1994), pp. 59–79.Google Scholar
[15]Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. (1965), p. 231.Google Scholar
[16]Grothendieck, A.Cohomologie l-adique et functions l. In Groupes de monodromie en géométrie algébrique. I (Springer-Verlag, Berlin, 1977), pp. 351–371. Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Exposé VIII. Dirigé par A. Grothendieck. Lecture Notes in Mathematics, vol. 589.Google Scholar
[17]Hartshorne, R.Algebraic Geometry. Graduate Texts in Math. vol. 52 (Springer-Verlag, New York, 1977).CrossRefGoogle Scholar
[18]Ruipérez, D. Hernández, Martín, A. C. López, Gómez, D. Sánchez and Prieto, C. Tejero Moduli spaces of semistable sheaves on singular genus 1 curves. Int. Math. Res. Not. IMRN (2009), pp. 4428–4462.Google Scholar
[19]Ruipérez, D. Hernández, Martín, A. C. López and de Salas, F. SanchoFourier-Mukai transform for Gorenstein schemes. Adv. Math. 211 (2007), pp. 594–620.CrossRefGoogle Scholar
[20]Ruipérez, D. Hernández, Martín, A. C. López and de Salas, F. SanchoRelative integral functors for singular fibrations and singular partners. J. Eur. Math. Soc. 11 (2009), pp. 597–625.CrossRefGoogle Scholar
[21]Ruipérez, D. Hernández and Porras, J. M. MuñozStable sheaves on elliptic fibrations. J. Geom. Phys. 43 (2002), pp. 163–183.CrossRefGoogle Scholar
[22]Hille, L. and Van den Bergh, M. D.Fourier-Mukai transforms. In Handbook on tilting theory. London Math. Soc. Lecture Note Series vol. 332 (Cambridge University Press, 2007).Google Scholar
[23]Huybrechts, D.Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs (The Clarendon Press – Oxford University Press, 2006.CrossRefGoogle Scholar
[24]Huybrechts, D., Macrì, E. and Stellari, P.Derived equivalences of K3 surfaces and orientation. Duke Math. J. 149 (2009), pp. 461–507.CrossRefGoogle Scholar
[25]Kodaira, K.On compact analytic surfaces. II, III. Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), pp. 1–40.CrossRefGoogle Scholar
[26]Kuznetsov, A. Homological projective duality. Publ. Math. Inst. Hautes Études Sci. (2007), pp. 157–220.Google Scholar
[27]Lenstra, H. W. Jr., Oort, F. and Zarhin, Y. G.Abelian subvarieties. J. Algebra. 180 (1996), pp. 513–516.CrossRefGoogle Scholar
[28]Lunts, V. A. and Orlov, D. O.Uniqueness of enhancement for triangulated categories. J. Amer. Math. Soc. 23 (2010), pp. 853–908.CrossRefGoogle Scholar
[29]Miranda, R.The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. Doctorate in Mathematical Research (ETS Editrice, Pisa, 1989).Google Scholar
[30]Mukai, S.Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18 (1978), pp. 239–272.Google Scholar
[31]Mukai, S.Fourier functor and its application to the moduli of bundles on an abelian variety. In Algebraic Geometry (Sendai, 1985, Adv. Stud. Pure Math., vol. 10 (North-Holland, Amsterdam, 1987), pp. 515–550.Google Scholar
[32]Mumford, D., Fogarty, J. and Kirwan, F.Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2). Results in Mathematics and Related Areas (2) vol. 34 (Springer-Verlag, Berlin, third ed., 1994).Google Scholar
[33]Orlov, D. O.Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York). 84 (1997), pp. 1361–1381. Algebraic geometry, 7.CrossRefGoogle Scholar
[34]Orlov, D. O.Derived categories of coherent sheaves on abelian varieties and equivalences between them. Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), pp. 131–158.Google Scholar
[35]Polishchuk, A.Symplectic biextensions and a generalization of the Fourier-Mukai transform. Math. Res. Lett. 3 (1996), pp. 813–828.CrossRefGoogle Scholar
[36]Polishchuk, A.Analogue of Weil representation for abelian schemes. J. Reine Angew. Math. 543 (2002), pp. 1–37.CrossRefGoogle Scholar
[37]Raynaud, M.Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Math. vol. 119 (Springer-Verlag, Berlin, 1970).Google Scholar
[38]Sancho de Salas, C. and de Salas, F. Sancho Reconstructing schemes from the derived category. Preprint, 2010.Google Scholar
[39]Seidel, P. and Thomas, R.Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108 (2001), pp. 37–108.CrossRefGoogle Scholar
[40]Spaltenstein, N.Resolutions of unbounded complexes. Compositio Math. 65 (1988), pp. 121–154.Google Scholar