Mathematische Annalen

, Volume 357, Issue 2, pp 743–759 | Cite as

Derived categories of Burniat surfaces and exceptional collections

  • Valery Alexeev
  • Dmitri Orlov


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 \(K_0(\mathcal{A })=\mathbb{Z }_2^6\).


Pezzo Surface Coherent Sheave Ample Line Bundle Hochschild Homology Exceptional Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Algebraic Geometry SectionSteklov Mathematical Institute RASMoscowRussia

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