Exceptional collections on toric manifolds

Exceptional collections serve as a fantastic tool to study and describe the structure of D^b(X) of certain algebraic varieties (hence the name “exceptional”) . The first example of such (full strongly exceptional) collections was given by Beilinson in his seminal paper [B] in the case of X=ℙⁿ. The collections given by Beilinson are ℰ={O(i):i=1,...,n} and ℰ’={Ωⁱ(i):i=1,...,n}. In general the definition is the following:

Definition: Let D^b(X) be the derived category of coherent sheaves on an algebraic manifold X.

(1) An object E∈D^b(X) is said to be exceptional if Hom(E,E)=ℂ and Extⁱ(E,E)=0 for i≠0.

(2) An ordered collection ℰ={E,...,E}⊂D(X) of exceptional objects is said to be an exceptional collection if Extⁱ(E_j,E_k)=0

for all j

(3) An exceptional collection is strongly exceptional if also Extⁱ(E_j,E_k)=0 for all j≤k and i≠0.

(4) A collection ℰ is called full if it generates the derived category D^b(X) as a triangulated category.

Given a collection ℰ={E₁,...,E_N} one associates to it the algebra A=Hom(T,T) where T=E₁⊕...⊕E_N. The importance of full strongly exceptional collections is due to the following property:

 Theorem (see [B,Bo1]): Let ℰ={E₁,...,E_N}⊂D^b(X) be a full strongly exceptional collection. Then there is an equivalence of categories D^b(X)=D^b(A).

Since the first examples were discovered by Beilinson, many further examples of full strongly exceptional collections were found on other algebraic manifolds, by various authors. A class of manifolds on which exceptional collections were extensively studied is the class of toric manifolds. In [K] A. King asked the following question: which toric manifolds X admit a full strongly exceptional collection of line bundles in Pic(X). On the one hand, many examples of full strongly exceptional collections of line bundles on toric manifolds were found, a partial list is [Bo3,BH,BT,CMR1,CMR2,CMR3,CMR4,CDRMR,Ka,LM,P]. On the other hand, Hille and Perling found an example of a toric surface which cannot admit such a collection [HP] a Fano class of examples was recently found by Efimov in [E]. Our interest in full strongly exceptional collections of line bundles comes from the fact that they seem to naturally arise in the context of Landau-Ginzburg systems and homological mirror symmetry for toric manifolds.

Landau-Ginzburg systems

Homological mirror symmetry

References

[B] A. Beilinson. The derived category of coherent sheaves on . Selected translations. Selecta Math. Soviet. 3 (1983/84), no. 3, 233-237.

[Bo1] A. Bondal. Helices, representations of quivers and Koszul algebras. Helices and vector bundles, 75-95, London Math. Soc. Lecture Note Ser., 148, Cambridge Univ. Press, Cambridge, 1990.

[Bo2] A. Bondal. Representations of associative algebras and coherent sheaves. Math. USSR-Izv. 34 (1990), no. 1, 23-42.

[Bo3] A. Bondal. Derived categories of toric varieties. Obervolfach reports, 3 (1), 284-286, 2006.

[BH] L. Borisov, Z. Hua. On the conjecture of King for smooth toric Deligne-Mumford stacks. Adv. Math. 221 (2009), no. 1, 277-301.

[BT] A. Bernardi, S. Tirabassi. Derived categories of toric Fano 3-folds via the Frobenius morphism. Matematiche (Catania) 64 (2009), no. 2, 117-154.

[CMR1] L. Costa, R. M.Miro-Roig. Tilting sheaves on toric varieties. Math Z., 248 (2004), 849-865.

[CMR2] L. Costa, R. M. Miro-Roig. Derived categories of projective bundles. Proc. Amer. Math. Soc. 133 (2005), no. 9, 2533-2537.

[CMR3] L. Costa, R. M. Miro-Roig. Frobenius splitting and derived category of toric varieties. Illinois J. Math. 54 (2010), no. 2, 649-669.

[CMR4] L. Costa, R. M. Miro-Roig. Derived category of toric varieties with small Picard number. Cent. Eur. J. Math. 10 (2012), no. 4, 1280—1291.

[CDRMR] L. Costa, S. Di Rocco, R. M. Miró-Roig. Derived category of fibrations. Math. Res. Lett. 18 (2011), no. 3, 425-432.

[E] A. Efimov. Maximal lengths of exceptional collections of line bundles. J. London Math. Soc., 90:2 (2014), 350—372.

[HP] L. Hille, M. Perling. A counterexample to King's conjecture. Compos. Math. 142 (2006), no. 6, 1507-1521.

[Ka] Y. Kawamata. Derived categories of toric varieties. Michigan Math. J. 54 (2006), no. 3, 517-535.

[K] A. King. Tilting bundles on some rational surfaces. preprint.

[O] D. O. Orlov. Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. RAN. Ser. Mat., 1992, Volume 56, Issue 4, 852—862.

[LM] M. Lason, M. Michalek. On the full, strongly exceptional collections on toric varieties with Picard number three. Collect. Math. 62 (2011), no. 3, 275-296.

[P] M. Perling. Some Quivers Describing the Derived Category of the Toric del Pezzos. Preprint 2003.