Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1569-1592 |
| Number of pages | 24 |
| Journal | Journal of the European Mathematical Society |
| Volume | 17 |
| Issue number | 7 |
| DOIs | |
| State | Published - 2015 |
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Keywords
- Barlow surfaces
- Derived categories
- Exceptional collections
- Hochschild homology
- Semiorthogonal decompositions
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Determinantal Barlow surfaces and phantom categories. / Böhning, Christian; Graf Von Bothmer, Hans Christian; Katzarkov, Ludmil; Sosna, Pawel.
In: Journal of the European Mathematical Society, Vol. 17, No. 7, 2015, p. 1569-1592.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Determinantal Barlow surfaces and phantom categories
AU - Böhning, Christian
AU - Graf Von Bothmer, Hans Christian
AU - Katzarkov, Ludmil
AU - Sosna, Pawel
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Barlow surfaces
KW - Derived categories
KW - Exceptional collections
KW - Hochschild homology
KW - Semiorthogonal decompositions
UR - http://www.scopus.com/inward/record.url?scp=84932164956&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84932164956&partnerID=8YFLogxK
U2 - 10.4171/JEMS/539
DO - 10.4171/JEMS/539
M3 - Article
AN - SCOPUS:84932164956
VL - 17
SP - 1569
EP - 1592
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
SN - 1435-9855
IS - 7
ER -