Derived categories of Burniat surfaces and exceptional collections

By Valery Alexeev, Dmitri Orlov. 2012

56 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/92d6d338402eb56a4631641e6f726ff8887f6edb

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Citations in the corpus

Exceptional collections on isotropic Grassmannians (2011, background): “G/P is full. The more general Nonvanishing Conjecture of [K09] stating that every exceptional collection of expected length is full turned out to be false — counterexamples were constructed in [BBS], [AO], [GS], [BBKS]. Nevertheless, we believe that the conjecture is still true for homogeneous space”
Exceptional collections of line bundles on the Beauville surface (2012, methodology): “Proposition 3.7. All of our exceptional collections in question are non-strict. Moreover in all of them both Ext1 and Ext2 are present unlike the case of the Burniat surfaces where only Ext2 appears ([AO], Lemma 4.8). We also note that unlike the case of Burniat surfaces the exceptional collections ”
Semiorthogonal decompositions in algebraic geometry (2014, background): “In fact, for many surfaces of general type with pg = q = 0 (the classical Godeaux surface, the Beauville surface, the Burniat surfaces, the determinantal Barlow surface, some fake projective planes) exceptional collections of length equal to the rank of the Grothendieck group have been constructed i”
On the derived category of the classical Godeaux surface (2012, methodology): “The argument is inspired by the proof of rigidity for the exceptional sequences similar to ours in the recent preprint [1] by Alexeev and Orlov.”