Pawel Sosna

On the derived category of the classical Godeaux surface

Abstract We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z / 5 Z -quotient of the Fermat quintic surface in P 3 . This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z 11 ⊕ Z / 5 Z . In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E 8 . We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.

Determinantal Barlow surfaces and phantom categories

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 Kuznetsovs 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.

On the Jordan–Hölder property for geometric derived categories

Abstract We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan–Holder property. More precisely, there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. Assuming the Noetherian property for semiorthogonal decompositions, one can define, following Kuznetsov, the Clemens–Griffiths component CG ( D ) for each fixed maximal decomposition D . We then show that D b ( X ) has two different maximal decompositions for which the Clemens–Griffiths components differ. Moreover, we produce examples of rational fourfolds whose derived categories also violate the Jordan–Holder property.

Linearisations of triangulated categories with respect to finite group actions

Given an action of a finite group on a triangulated category, we investigate under which conditions one can construct a linearised triangulated category using DG-enhancements. In particular, if the group is a finite group of automorphisms of a smooth projective variety and the category is the bounded derived category of coherent sheaves, then our construction produces the bounded derived category of coherent sheaves on the smooth quotient variety resp. stack. We also consider the action given by the tensor product with a torsion canonical bundle and the action of a finite group on the category generated by a spherical object.

Scalar Extensions of Triangulated Categories

Given a triangulated category

Equivalences of equivariant derived categories

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Stability conditions under change of base field

over a field K and a field extension L/K, we investigate how one can construct a triangulated category

Some properties of dynamical degrees with a view towards cubic fourfolds

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Fourier-Mukai Partners of Canonical Covers of Bielliptic and Enriques Surfaces

over L. Our approach produces the derived category of the base change scheme XL if

Derived equivalent conjugate K3 surfaces

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