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=ℙ​n. The collections given by Beilinson are ℰ={O(i):i=1,...,n} and ℰ’={Ω​i(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 Ext​i(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 Ext​i(E​j,E​k)=0
                  for all j
             (3) An exceptional collection is strongly exceptional if also Ext​i​(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={E​1,...,E​N} one associates to it the algebra A=Hom(T,T)  where T=E​1⊕...⊕E​N. The importance of full strongly exceptional collections is due to the following property:

​Theorem ​(see [B,Bo1]):​ Let ℰ={E​1,...,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

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