Journal of the European Mathematical Society

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Volume 17, Issue 7, 2015, pp. 1569–1592
DOI: 10.4171/JEMS/539

Published online: 2015-06-23

Determinantal Barlow surfaces and phantom categories

Christian Böhning[1], Hans-Christian Graf von Bothmer[2], Ludmil Katzarkov[3] and Pawel Sosna[4]

(1) Universität Hamburg, Germany
(2) Universität Hamburg, Germany
(3) University of Miami, Coral Gables, USA
(4) Universität Hamburg, Germany

We prove that the bounded derived category of the surface $S$ constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of $S$ in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov’s results on heights of exceptional sequences, we also show that the sequence on $S$ itself is not full and its (left or right) orthogonal complement is also a phantom category.

Keywords: Derived categories, exceptional collections, semiorthogonal decompositions, Hochschild homology, Barlow surfaces

Böhning Christian, Graf von Bothmer Hans-Christian, Katzarkov Ludmil, Sosna Pawel: Determinantal Barlow surfaces and phantom categories. J. Eur. Math. Soc. 17 (2015), 1569-1592. doi: 10.4171/JEMS/539