Derived Categories of Toric Varieties

By Y. Kawamata. 2005

205 citations recorded by the source corpus.

Theorem 1.1. Let X be a projective toric variety with at most quotient singularities, let B be an invariant Q-divisor whose coefficients belong to the set { r−1 r ; r ∈ Z>0 } , and let X be the smooth Deligne–Mumford stack associated to the pair (X,B) as in [12]. Then the bounded derived category of coherent sheaves Db(Coh(X )) has a complete exceptional collection consisting of sheaves. An object of a triangulated category a ∈ T is called exceptional if

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f1ffe7379ba0a74917c31402b0519a13ecbcf66

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Citations in the corpus

The derived category of a GIT quotient (2012, methodology): “The semiorthogonal decomposition in this theorem can be refined further using ideas of Kawamata [17], and Ballard, Favero, Katzarkov [4].”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, unknown): “We thank M. Abouzaid, A. Bondal and P. Seidel for explaining relevant aspects of their work, and D. Nadler for helpful suggestions.”
Projective toric varieties as fine moduli spaces of quiver representations (2006, background): “[20] proves that every toric variety has a complete exceptional collection of coherent sheaves and, in contrast, [18] exhibits a smooth toric surface that does not have a complete strong exceptional collection of line bundles.”
Brane tilings and their applications (2008, background): “For the toric case we are now going to consider, the existence of full exceptional collections was shown by Kawamata [237].”
On the conjecture of King for smooth toric Deligne-Mumford stacks (2008, background): “The case of Picard number one has already been settled in [11], but we have treated it here nonetheless, to give a unified picture of our approach.”
Quantum Cohomology and Periods (2011, methodology): “This holds true in our case because X does not have generic stabilizers and we can apply the result of Kawamata [41] that the derived category Db(X ) of coherent sheaves has a full exceptional collection {E}Ni=1.”
A categorification of Morelli’s theorem (2010, background): “This kind of “coherent-constructible correspondence” was first observed by Bondal [4].”
Semiorthogonal decompositions in algebraic geometry (2014, background): “Full exceptional collections on smooth toric varieties (and stacks) were constructed by Kawamata [Kaw06].”
Constructible sheaves and the Fukaya category (2006, result): “First, the announced results of Bondal and Bondal-Ruan [4] relate the derived categories of coherent sheaves on toric Fano varieties with the Fukaya-Seidel category on the Landau-Ginzburg side.”
The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, background): “The nonequivariant CCC has been studied by Bondal [8], and more recently by the third author [51]. diagram: (2) PerfT (XΣ) κ // τ ''PP PP PP PP PP PP Shcc(MR; ΛΣ) µ F (T ∗MR; ΛΣ) where F = TrFuk is the triangulated envelope of the Fukaya category.”