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Derived categories of Burniat surfaces and exceptional collections

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We construct an exceptional collection \(\varUpsilon \) of maximal possible length 6 on any of the Burniat surfaces with \(K_X^2=6\), a 4-dimensional family of surfaces of general type with \(p_g=q=0\). We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement \(\mathcal{A }\) of \(\varUpsilon \) is an “almost phantom” category: it has trivial Hochschild homology, and \(K_0(\mathcal{A })=\mathbb{Z }_2^6\).

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Acknowledgments

We thank Rita Pardini for providing us with a proof of Lemma 2 and for helpful comments. We also would like to thank the University of Vienna and Ludmil Katzarkov for organizing a workshop on Birational geometry and Mirror symmetry during which this project was started.The first author was supported by the NSF under DMS-1200726. The second author was partially supported by RFBR grants 10-01-93113, 11-01-00336, 11-01-00568, NSh Grant 4713.2010.1, by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30605, USA

    Valery Alexeev

  2. Algebraic Geometry Section, Steklov Mathematical Institute RAS, Gubkin str. 8, Moscow, 119991, Russia

    Dmitri Orlov

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  1. Valery Alexeev
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  2. Dmitri Orlov
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Correspondence to Dmitri Orlov.

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Alexeev, V., Orlov, D. Derived categories of Burniat surfaces and exceptional collections. Math. Ann. 357, 743–759 (2013). https://doi.org/10.1007/s00208-013-0917-2

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