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Derived categories of Burniat surfaces and exceptional collections. (English) Zbl 1282.14030

A Burniat surface is a smooth surface of general type \(X\) over an algebraically closed field \(\mathbb{K}\) of characteristic \(\neq 2\) satisfying \(p_g=H^2(X,\mathcal{O}_X)=0=H^1(X,\mathcal{O}_X)=q\) and \(K_X^2=6\). These surfaces form a \(4\)-dimensional family. Most of these surfaces (over \(\mathbb{C}\) all of them) can be conveniently described as a Galois \(\mathbb{Z}^2_2\)-cover of \(\text{Bl}_3\mathbb{P}^2\), the blow-up of the projective plane in three points. The main result of the paper describes the structure of \(D^b(X)\), the bounded derived category of coherent sheaves on \(X\). To be more precise, recall that an object \(E\in D^b(X)\) is called exceptional if \(\text{Hom}(E,E[l])=\mathbb{K}\) for \(l=0\) and \(\text{Hom}(E,E[l])=0\) for \(l\neq 0\). For example, any line bundle on \(X\) is exceptional. An exceptional collection \((E_1,\ldots,E_k)\) is an ordered sequence of exceptional objects satisfying \(\text{Hom}(E_m,E_n[l])=0\) for \(m>n\) and for all \(l\). The authors prove that on any Burniat surface \(X\) which is a Galois cover there exists an exceptional collection of length \(6\) consisting of line bundles \(L_i\). This implies that the orthogonal complement \(\mathcal{T}\) of this collection, that is the category of objects \(F\) satisfying \(\text{Hom}(F,L_i[l])=0\) for all \(i\) and all \(l\), has trivial Hochschild homology and finite Grothendieck group.
The paper is organized as follows. In Section 2 the authors study the Picard group of a Burniat surface via its close connection to \(\text{Bl}_3\mathbb{P}^2\) mentioned above. Section 3 recalls some well-known results concerning exceptional collections on \(\text{Bl}_3\mathbb{P}^2\). In Section 4 the authors construct exceptional sequences of maximal possible length \(6\) using the results from Section 2. More precisely, they show that these sequences split into so-called blocks, similar to what happens on \(\text{Bl}_3\mathbb{P}^2\). Furthermore, denoting \(\bigoplus_i L_i\) by \(T\), the authors calculate the DG-algebra of endomorphisms of \(T\) and prove that it is formal. The latter statement implies, in particular, that the category generated by the exceptional collection does not change when the surface is varied. The paper concludes with some remarks in Section 5.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J29 Surfaces of general type
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[1] Alexeev, V., Pardini, R.: Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, p. 26 (2009, Preprint). arXiv:0901.4431 · Zbl 0329.14019
[2] Bauer, I., Catanese, F.: Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces. Classification of algebraic varieties, EMS Series of Congress Report, European Mathematical Society, Zürich, pp. 49-76 (2011) · Zbl 1264.14052
[3] Böhning, C., Graf von Bothmer, H-C., Sosna, P.: On the derived category of the classical Godeaux surface (2012, Preprint). arXiv:1206.1830v1 · Zbl 1299.14015
[4] Böhning, C., Graf von Bothmer, H-C., Katzarkov, L., Sosna, P.: Determinantal Barlow surfaces and phantom categories (2012, Preprint). arXiv:1210.0343 · Zbl 1323.14014
[5] Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, 2nd edn. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer, Berlin (2004) · Zbl 1036.14016
[6] Bondal, A., Kapranov, M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183-1205, 1337 (1989) · Zbl 0703.14011
[7] Bondal, A., Kapranov, M.: Enhanced triangulated categories. Mat. Sb. 181(5), 669-683 (1990) · Zbl 0719.18005
[8] Bondal, A., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125(3), 327-344 (2001) · Zbl 0994.18007
[9] Burniat, P.: Sur les surfaces de genre \[P_{12}{{\>}}1\]. Ann. Mat. Pura Appl. (4) 71, 1-24 (1966) · Zbl 0144.20203
[10] Diemer, C., Katzarkov, L., Kerr, G.: Compactifications of spaces of Landau-Ginzburg models (2012, Preprint). arXiv:1207.0042v1 · Zbl 1277.14015
[11] Galkin, S., Shinder, E.: Exceptional collections of line bundles on the Beauville surface (2012, Preprint). arXiv:1210.3339 · Zbl 1408.14068
[12] Gorchinskiy, S., Orlov, D.: Geometric Phantom Categories. Publications IHES (2013). arXiv: 1209.6183 · Zbl 1285.14018
[13] Inose, H., Mizukami, M.: Rational equivalence of \[0\]-cycles on some surfaces of general type with \[p_g=0\]. Math. Ann. 244(3), 205-217 (1979) · Zbl 0444.14006
[14] Inoue, M.: Some new surfaces of general type. Tokyo J. Math. 17(2), 295-319 (1994) · Zbl 0836.14020
[15] Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63-102 (1994) · Zbl 0799.18007
[16] Keller, B.: Introduction to \[A\]-infinity algebras and modules. Homol. Homotopy Appl. 3(1), 1-35 (2001) · Zbl 0989.18009
[17] Keller, B.: On differential graded categories. International Congress of Mathematicians, vol. II, European Mathematical Society, Zürich, pp. 151-190 (2006) · Zbl 1140.18008
[18] Karpov, B.V., Nogin, D.Yu.: Three-block exceptional sets on del Pezzo surfaces. Izv. Ross. Akad. Nauk Ser. Mat. 62(3), 3-38 (1998) · Zbl 0949.14026
[19] Kuleshov, S.A., Orlov, D.O.: Exceptional sheaves on Del Pezzo surfaces. Izv. Ross. Akad. Nauk Ser. Mat. 58(3), 53-87 (1994)
[20] Kuznetsov, A.: Hochschild homology and semiorthogonal decompositions (2009, Preprint). arXiv: 0904.4330v1 · Zbl 0799.18007
[21] Lefevre, K.: Sur les \[A_{\infty }\]-catégories. Ph.D. thesis, Université Paris 7 (2002)
[22] Mendes Lopes, M., Pardini, R.: A connected component of the moduli space of surfaces with \[p_g=0\]. Topology 40(5), 977-991 (2001) · Zbl 1072.14522
[23] Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 852-862 (1992) · Zbl 0798.14007
[24] Orlov, D.: Remarks on generators and dimensions of triangulated categories. Mosc. Math. J. 9(1), 153-159 (2009)
[25] Pardini, R.: Abelian covers of algebraic varieties. J. Reine Angew. Math. 417, 191-213 (1991) · Zbl 0721.14009
[26] Peters, C.A.M.: On certain examples of surfaces with \[p_g=0\] due to Burniat. Nagoya Math. J. 66, 109-119 (1977) · Zbl 0329.14019
[27] Seidel, P.: Fukaya Categories and Picard-Lefschetz Theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) · Zbl 1159.53001
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