An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves

Journal of Geometry and Physics

2017

This paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors.

Section: Bad News Again: Lifting Functorsmentioning

confidence: 99%

“…Unfortunately, not only the proof of this result, but even the definition of the functor is rather involved, as it uses sophisticated techniques from deformation theory. So we will limit ourselves to give a rough idea of their construction, inviting the interested reader to consult directly [54] for more details.…”

Section: Bad News Again: Lifting Functorsmentioning

confidence: 99%

A tour about existence and uniqueness of dg enhancements and lifts

Canonaco

Journal of Geometry and Physics

2017

“…These functors are ubiquitous in algebraic geometry (see [7] for a survey on the subject) and for a long while it was believed by some people that all exact functors between D b (X 1 ) and D b (X 2 ), with X i a smooth projective scheme, had to be of Fourier-Mukai type. A beautiful counterexample by Rizzardo and Van den Bergh [34] showed this expectation to be false. Moreover, if X 1 and X 2 are not smooth projective it is not even clear if the celebrated result of Orlov [31] asserting that all exact equivalences between D b (X 1 ) and D b (X 2 ) are of Fourier-Mukai type holds true.…”

Section: 2mentioning

confidence: 99%

Uniqueness of dg enhancements for the derived category of a Grothendieck category

Canonaco

2018

J. Eur. Math. Soc.

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.

“…Remark 2.19. Not all functors D 1 → D 2 are of Fourier-Mukai type (see [RvdB14,Vol16]). Nevertheless, fully faithful functors are expected to be of Fourier-Mukai type (as they are in the commutative case, by Orlov's Representability Theorem [Orl97]).…”

Section: 2mentioning

confidence: 99%

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

Macrì

Lecture Notes of the Unione Matematica Italiana

2019

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