On a Relative Fourier–Mukai Transform on Genus One Fibrations

Burban, Igor; Kreussler, Bernd

doi:10.1007/s00229-006-0013-y

Cited by 16 publications

(22 citation statements)

References 27 publications

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“…In the third and fourth sections, using the relative Jacobian, we adapt the construction of Cȃldȃraru, [Cȃl02], to our case, obtaining a twisted Fourier-Mukai transform. Similar results were obtained in different settings by O.Ben-Bassat [BB09] and I.Burban and B.Kreussler [BK06]. In the fifths section using this transform and the associated spectral cover we prove that the moduli space of rank n, relatively semi-stable vector bundles is corepresented by the relative Douady space of length n and relative dimension 0 subspaces of the relative Jacobian, see theorem 5.…”

Section: Introductionsupporting

confidence: 80%

Vector bundles on non-Kaehler elliptic principal bundles

Brînzănescu

Halanay²,

Trautmann³

2013

Ann. inst. Fourier

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Abstract. We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted FourierMukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional nonKähler elliptic principal bundle over a primary Kodaira surface are computed.

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Section: Introductionsupporting

confidence: 80%

Vector bundles on non-Kaehler elliptic principal bundles

Brînzănescu

Halanay²,

Trautmann³

2013

Ann. inst. Fourier

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“…The conjectures made in [19] form the main geometric motivation for this paper, although the methods of proof will be closer to those found in [14]. More recently, these kinds of results have been used and expanded upon by other authors; for example see [11], [13], and [31]. An alternative, and in many ways stronger, version of the main theorem can be found as Theorem 5.12.…”

Section: Oren Ben-bassatmentioning

confidence: 96%

Twisting derived equivalences

Ben-Bassat

2009

Trans. Amer. Math. Soc.

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Abstract. We introduce a new method for "twisting" relative equivalences of derived categories of sheaves on two spaces over the same base. The first aspect of this is that the derived categories of sheaves on the spaces are twisted. They become derived categories of sheaves on gerbes living over spaces that are locally (on the base) isomorphic to the original spaces. Secondly, this is done in a compatible way so that the equivalence is maintained. We apply this method by proving the conjectures of Donagi and Pantev on dualities between gerbes on genus-one fibrations and comment on other applications to families of higher genus curves. We also include a related conjecture in Mirror Symmetry.

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“…For every closed point s ∈ S, the absolute functor s = [11]. They proved a version of Proposition 2.16 in the case when the base field is of characteristic zero, S and X are reduced, X is connected, the fibration p : X → S has only integral fibres and it has a section taking values in the smooth locus of X.…”

Section: Restriction To Fibres: a Criterion For Equivalencementioning

confidence: 99%

Relative integral functors for singular fibrations and singular partners

Ruipérez¹,

Martín²,

Salas³

2009

J. Eur. Math. Soc.

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Abstract. We study relative integral functors for singular schemes and characterise those which preserve boundedness and those which have integral right adjoints. We prove that a relative integral functor is an equivalence if and only if its restriction to every fibre is an equivalence. This allows us to construct a non-trivial auto-equivalence of the derived category of an arbitrary genus one fibration with no conditions on either the base or the total space and getting rid of the usual assumption of irreducibility of the fibres. We also extend to Cohen-Macaulay schemes the criterion of Bondal and Orlov for an integral functor to be fully faithful in characteristic zero and give a different criterion which is valid in arbitrary characteristic. Finally, we prove that for projective schemes both the Cohen-Macaulay and the Gorenstein conditions are invariant under Fourier-Mukai functors.

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On a Relative Fourier–Mukai Transform on Genus One Fibrations

Cited by 16 publications

References 27 publications

Vector bundles on non-Kaehler elliptic principal bundles

Vector bundles on non-Kaehler elliptic principal bundles

Twisting derived equivalences

Relative integral functors for singular fibrations and singular partners

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