Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds

Maciocia, Antony; Piyaratne, Dulip

doi:10.14231/ag-2015-012

Cited by 58 publications

(82 citation statements)

References 27 publications

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“…Our first main result is the following, generalizing the result of [29,30] for the case when X has Picard rank one:…”

Section: Stability Conditions On Threefolds Via a Conjectural Bogomolmentioning

confidence: 78%

“…Until recent work by Maciocia and Piyaratne [29,30], they were only known to exist on threefolds whose derived category admits a full exceptional collection. Possible applications of stability conditions range from modularity properties of generating functions of Donaldson-Thomas invariants [43,45] to Reider-type theorems for adjoint linear series [6].…”

Section: Stability Conditions On Threefolds Via a Conjectural Bogomolmentioning

confidence: 99%

“…As indicated above, the first breakthrough towards constructing stability conditions on threefolds (without using exceptional collections) is due to Maciocia and Piyaratne, who proved Theorem 1.1 in the case of principally polarized abelian varieties of Picard rank one in [29,30]. Their method is based on an extensive analysis of the behavior of tilt-stability with respect to FourierMukai transforms; in addition to constructing stability conditions, they show their invariance under Fourier-Mukai transforms.…”

Section: Related Workmentioning

confidence: 99%

“…For example, a crucial step in their arguments uses restriction to divisors and curves to control a certain cohomology sheaf of the Fourier-Mukai transform of E, see the proof of [29,Proposition 4.15]; in Sect. 7 we use restriction of divisors explicitly and to curves implicitly (when we use Theorem 7.2) to control global sections of pull-backs of E.…”

Section: Related Workmentioning

confidence: 99%

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The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

2016

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We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps:1. We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involv- ing the third Chern character of "tilt-stable" two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. 2. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. 3. We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne.Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

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“…Our first main result is the following, generalizing the result of [29,30] for the case when X has Picard rank one:…”

Section: Stability Conditions On Threefolds Via a Conjectural Bogomolmentioning

confidence: 78%

Section: Stability Conditions On Threefolds Via a Conjectural Bogomolmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

2016

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“…A similar argument was then successfully applied to the smooth quadric hypersurface in P 4 in [Sch14]. The case of abelian threefolds was independently proved in [MP15,MP16] and [BMS14]. Moreover, as pointed out in [BMS14], this also implies the case ofétale quotients of abelian threefolds and gives the existence of Bridgeland stability condition on orbifold quotients of abelian threefolds (this includes examples of Calabi-Yau threefolds which are simply-connected).…”

Section: Introductionmentioning

confidence: 79%

Lectures on Bridgeland Stability

Macrì

Schmidt

2017

Lecture Notes of the Unione Matematica Italiana

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Abstract. In these lecture notes we give an introduction to Bridgeland stability conditions on smooth complex projective varieties with a particular focus on the case of surfaces. This includes basic definitions of stability conditions on derived categories, basics on moduli spaces of stable objects and variation of stability. These notes originated from lecture series by the first author at the summer school Recent advances in algebraic and arithmetic geometry, Siena, Italy, August 24-28, 2015 and at the school Moduli of Curves, CIMAT, Guanajuato, Mexico, February 22 -March 4, 2016.

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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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Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds

Cited by 58 publications

References 27 publications

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

Lectures on Bridgeland Stability

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

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