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

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Birational Geometry of Hypersurfaces

Abstract

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 Birational Geometry of Hypersurfaces, Palazzo Feltrinelli - Gargnano del Garda (Italy), March 19–23, 2018.

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Notes

  1. 1.

    A triangulated category \(\mathcal {D}\) is proper over \(\mathbb K\) if, for all \(F,G\in \mathcal {D}\), \(\dim _{\mathbb K} (\oplus _p \operatorname {\mathrm {Hom}}_{\mathcal {D}}(F,G[p]))<+\infty \).

  2. 2.

    Note that in [136, Definition 4.3] it is used the terminology geometric noncommutative scheme for what we call non-commutative smooth projective variety.

  3. 3.

    This is different from the definition which appears in [96, Equation (10)], but in our context of smooth projective varieties it is equivalent (see also the beginning of the proof of [96, Theorem 5.8]).

  4. 4.

    Hochschild (co)homology for schemes was originally defined and studied in [112, 153, 161].

  5. 5.

    In [98] the functor O is defined on the whole derived category and it is called rotation functor.

  6. 6.

    If n = 2, then X is a K3 surface. If n is odd, everything goes through in the same way, but the Kuznetsov component is an Enriques-type category, with S 2 = [4].

  7. 7.

    If n = 4, we have d = 2 and m = 4, for X ordinary, and d = 1 and m = 3, for X special; if n = 6, we have d = 1 and m = 5.

  8. 8.

    In our smooth setting, C-linearity simply means that each semiorthogonal factor is closed under tensorization by pull-backs of objects from .

  9. 9.

    We abuse notation and denote Z(v(A)) by Z(A). We use the identifications \(K(\mathcal {A})=K(\mathcal {D})\).

  10. 10.

    Note that such a line always exists as the family of lines in a smooth cubic fourfold are four-dimensional by [27]. On the other hand, such an hypersurface can contain only a finite number of planes.

  11. 11.

    This condition of openness and boundedness of stability should probably be assumed in the definition of Bridgeland stability condition; see indeed [23, 88].

  12. 12.

    The argument was also suggested to us by Claire Voisin.

  13. 13.

    It is actually enough to assume that the action of Φ on \(\widetilde {H}_{\mathrm {alg}}\) commutes with the action of the degree-shift functor.

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Acknowledgements

Our warmest thank goes to Alexander Kuznetsov: his many suggestions, corrections and observations helped us very much to improve the quality of this article. It is also our great pleasure to thank Arend Bayer, Andreas Hochenegger, Martí Lahoz, Howard Nuer, Alex Perry, Laura Pertusi, and Xiaolei Zhao for their insightful comments on the subject of these notes and for carefully reading a preliminary version of this paper. We are very grateful to Nick Addington, Enrico Arbarello and Daniel Huybrechts for many useful conversations and for patiently answering our questions, and to Amnon Yekutieli for pointing out the references [11] and [163]. We would also like to thank Andreas Hochenegger and Manfred Lehn for their collaboration in organizing the school these notes originated from, and the audience for many comments, critiques, and suggestions for improvements. Part of this paper was written while the second author was visiting Northeastern University. The warm hospitality is gratefully acknowledged.

The author “Emanuele Macrì” was partially supported by the NSF grant DMS-1700751. The author “Paolo Stellari” was partially supported by the ERC Consolidator Grant ERC-2017-CoG-771507-StabCondEn and by the research projects FIRB 2012 “Moduli Spaces and Their Applications” and PRIN 2015 “Geometria delle varietà proiettive”.

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Macrì, E., Stellari, P. (2019). Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces. In: Hochenegger, A., Lehn, M., Stellari, P. (eds) Birational Geometry of Hypersurfaces. Lecture Notes of the Unione Matematica Italiana, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-18638-8_6

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