On the Kuznetsov-Polishchuk conjecture

Fonarev, Anton

doi:10.1134/s0081543815060024

Cited by 11 publications

(11 citation statements)

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“…We conclude that . 6 Applying the induction hypothesis for l ′ = l − 2p we find that l − 2p = 2pq ′ + (p − 1). In particular, l = 2pq + (p − 1) for q = q ′ + 1.…”

Section: Isotropic Grassmannians Of Planesmentioning

confidence: 93%

“…(2) There are exactly three nonisomorphic equivariant irreducible Ulrich bundles on IGr (2,20): [6,6,6,6,6,6] (U ⊥ / U ), and U * (8) ⊗ Σ [8,8,6,6,4,4,2,2] (U ⊥ / U ).…”

Section: Isotropic Grassmannians Of Planesmentioning

confidence: 99%

“…On the other hand, according to our experience, it is very limiting to restrict oneself to the class of irreducible bundles. There seem to be equivariant bundles that are not irreducible, but still enjoy incredibly nice properties (see, e.g., [6]). It is also interesting to see if our results can help construct Ulrich bundles on other varieties, e.g.…”

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confidence: 99%

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Irreducible Ulrich Bundles on Isotropic Grassmannians

Fonarev¹

2016

MMJ

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We classify irreducible equivariant Ulrich vector bundles on isotropic Grassmannians. IntroductionUlrich bundles were introduced in [5] in order to study Chow forms. The notion of an Ulrich module appeared much earlier in commutative algebra. Ulrich himself gave a certain sharp upper bound on the minimal number of generators for a maximal Cohen-Macaulay module module over a Cohen-Macaulay homogeneous ring. Ulrich modules are precisely those for which the upper bound is attained (see [9]). There are several equivalent definitions of Ulrich bundles. Though we are going to use a less enlightning cohomological one, it might be motivating to formulate the most geometric and, to our taste, the easiest one. Given a d-dimensional projective variety X ⊂ P N , an Ulrich bundle on X is a vector bundle E such that π * E is a trivial bundle for a general linear projection π : X → P d . Further details may be found in [5].It was asked in [5] whether every projective variety admits an Ulrich bundle. So far the answer is known in few cases which include hypersurfaces and complete intersections [7], del Pezzo surfaces [5], and abelian surfaces [1].While it is not clear whether the answer to the general question is positive, in the case of rational homogeneous varieties one has a large test class of vector bundles, namely equivariant ones. In [4] the authors fully classified irreducible equivariant Ulrich vector bundles on Grassmannians over an algebraically closed field of characteristic zero. These results were further improved in [3], where most of partial flag varieties were treated. In the current paper we move in an orthogonal (or symplectic) direction and classify irreducible equivariant Ulrich bundles on isotropic Grassmannians, that is varieties of the form G/P, where G is a classical group of type B n , C n , or D n , and P is a maximal parabolic subgroup.The following meta theorem is the main result of the present paper.Theorem. The only isotropic Grassmannians admitting an equivariant irreducible Ulrich vector bundle are the symplectic Grassmannians of planes IGr(2, 2n) for n ≥ 2, odd and even-dimensional quadrics Q d , orthogonal Grassmannians of planes OGr(2, m) for m ≥ 4, even orthogonal Grassmannians of 3-spaces OGr(3, 4q + 6) for q ≥ 0, and OGr(4, 8). In each case the corresponding bundles are classified.The paper is organized as follows. In Section 2 we collect some preliminary definitions and provide a simple criterion for an equivariant irreducible vector bundle to be Ulrich. In Section 3 we treat all isotropic Grassmannians in type C n but maximal (Lagrangian) ones. Similarly, in sections 4 and 5 we deal with all but maximal Grassmannians in types B n and D n respectively. Finally, Section 6 is devoted to maximal Grassmannians, both symplectic and orthogonal.The author is gratful to the anonymous reviewer for careful reading of the original manuscript.

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“…We conclude that . 6 Applying the induction hypothesis for l ′ = l − 2p we find that l − 2p = 2pq ′ + (p − 1). In particular, l = 2pq + (p − 1) for q = q ′ + 1.…”

Section: Isotropic Grassmannians Of Planesmentioning

confidence: 93%

“…(2) There are exactly three nonisomorphic equivariant irreducible Ulrich bundles on IGr (2,20): [6,6,6,6,6,6] (U ⊥ / U ), and U * (8) ⊗ Σ [8,8,6,6,4,4,2,2] (U ⊥ / U ).…”

Section: Isotropic Grassmannians Of Planesmentioning

confidence: 99%

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Irreducible Ulrich Bundles on Isotropic Grassmannians

Fonarev¹

2016

MMJ

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“…For later use, we need a different exceptional collection, described more recently in [Fon15] (see also [KuPo16] for related results). We assume that gcd(k, n) = 1.…”

Section: Non-commutative Calabi-yau Varietiesmentioning

confidence: 99%

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

Macrì

Stellari

2019

Lecture Notes of the Unione Matematica Italiana

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We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds.These notes originated from the lecture series by the first author at the school on BirationalFrom the homological point of view, it was observed by Kuznetsov [Kuz10] that the derived category D b (W ) of a cubic fourfold W contains an admissible subcategory K u(W ) which is the right orthogonal of the category generated by the three line bundles O W , O W (H) and O W (2H). We will refer to K u(W ) as the Kuznetsov component of W . The category K u(W ) has the same homological properties of D b (S), for S a K3 surface: it is an indecomposable category with Serre functor which is the shift by 2 and the same Hochschild homology as D b (S). But, for W very general, there cannot be a K3 surface S with an equivalence K u(W ) ∼ = D b (S). This is the reason why we should think of Kuznetsov components as non-commutative K3 surfaces.The study of non-commutative varieties was started more than thirty years ago. Artin and Zhang [AZ84] investigated the case of non-commutative projective spaces (see also the book in preparation [Yek18]). In these notes we will follow closely the approach developed by Kuznetsov [Kuz10] and Huybrechts [Huy17]. One important feature is that the Kuznetzov component comes with a naturally associated lattice H(K u(W ), Z) with a weight-2 Hodge Non-commutative Calabi-Yau varietiesIn this section, we follow closely the presentation and main results in [Kuz15]; foundational references are also [BO95, BvdB03, Kuz07, Kuz14, Perr18a]. We start with a very short review on semiorthogonal decompositions and exceptional collections in Section 2.1. We then introduce the notion of non-commutative variety in Section 2.2 and study basic facts about Serre functors and Hochschild (co)homology. Finally, in Section 2.3 we sketch the

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“…Staircase complexes appeared in [Fon13], and were used to construct certain Lefschetz decompositions of the derived categories of the usual Grassmannians Gr(k, n). Their generalization later appeared in [Fon15], where the construction of Kuznetsov and Polishchuk was studied in type A.…”

Section: Introductionmentioning

confidence: 99%