Double spinor Calabi-Yau varieties

Manivel, Laurent

doi:10.46298/epiga.2019.volume3.3965

Cited by 14 publications

(4 citation statements)

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“…Similar phenomena can be observed if one replaces the Grassmannian G(2, 5) by the spinor variety S 10 [Ma18]. n = 6.…”

Section: Stratified Mukai Flopssupporting

confidence: 70%

Topics on the geometry of homogeneous spaces

Manivel

2020

Preprint

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This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent rôle. This selection is largely arbitrary and mainly reflects the interests of the author.Rational homogeneous varieties are very special projective varieties, which appear in a variety of circumstances as exhibiting extremal behavior. In the quite recent years, a series of very interesting examples of pairs (sometimes called Fourier-Mukai partners) of derived equivalent, but not isomorphic, and even non birationally equivalent manifolds have been discovered by several authors, starting from the special geometry of certain homogeneous spaces. We will not discuss derived categories and will not describe these derived equivalences: this would require more sophisticated tools and much ampler discussions. Neither will we say much about Homological Projective Duality, which can be considered as the unifying thread of all these apparently disparate examples. Our much more modest goal will be to describe their geometry, starting from the ambient homogeneous spaces.In order to do so, we will have to explain how one can approach homogeneous spaces just playing with Dynkin diagram, without knowing much about Lie theory. In particular we will explain how to describe the VMRT (variety of minimal rational tangents) of a generalized Grassmannian. This will show us how to compute the index of these varieties, remind us of their importance in the classification problem of Fano manifolds, in rigidity questions, and also, will explain their close relationships with prehomogeneous vector spaces.We will then consider vector bundles on homogeneous spaces, and use them to construct interesting birational transformations, including important types of flops: the Atiyah and Mukai flops, their stratified versions, also the Abuaf-Segal and Abuaf-Ueda flops; all these beautiful transformations are easily described in terms of homogenenous spaces. And introducing sections of the bundles involved, we will quickly arrive at several nice examples of Fourier-Mukai partners.We will also explain how the problem of finding crepant resolutions of orbit closures in prehomogeneous spaces is related to the construction of certain manifolds with trivial canonical class. This gives a unified perspective over classical constructions by Reid, Beauville-Donagi and Debarre-Voisin of abelian and hyperKähler varieties, naturally embedded into homogeneous spaces. The paper will close on a recent construction, made in a similar spirit, of a generalized Kummer fourfold from an alternating three-form in nine variables.

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“…Similar phenomena can be observed if one replaces the Grassmannian G(2, 5) by the spinor variety S 10 [Ma18]. n = 6.…”

Section: Stratified Mukai Flopssupporting

confidence: 70%

Topics on the geometry of homogeneous spaces

Manivel

2020

Preprint

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“…This is motivated by the first such example found in [Bor] [Mar16] and the other examples discovered in [IMOU] (the D-equivalence is shown later in [Kuz16]), [KS17], and [HL16] [IMOU16]. In addition, more supporting evidences have been discovered in the works [BCP17], [KR17], [Man17], and [KKM17]. In fact, all known examples are pairs of simply connected Calabi-Yau varieties with h 2,0 = 0 or K3 surfaces.…”

Section: Introductionmentioning

confidence: 85%

An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent

Okawa

2020

Math. Z.

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“…For , has two components and . These are called varieties of pure spinors (or spinor varieties), and these are isomorphic to each other [Ma]. For , and , the minuscule representations are familiar (see §6.5 in [BD]).…”

Section: Preliminariesmentioning

confidence: 99%

SOLUTIONS OF THE tt*-TODA EQUATIONS AND QUANTUM COHOMOLOGY OF MINUSCULE FLAG MANIFOLDS

Kaneko

2022

Nagoya Math. J.

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We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.

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Double spinor Calabi-Yau varieties

Cited by 14 publications

References 20 publications

Topics on the geometry of homogeneous spaces

Topics on the geometry of homogeneous spaces

An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent

SOLUTIONS OF THE tt*-TODA EQUATIONS AND QUANTUM COHOMOLOGY OF MINUSCULE FLAG MANIFOLDS

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