Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Tilting Sheaves on toric varieties by Rosa M. Miró-Roig, Universitat de Barcelona Abstract: (Joint work with L. Costa) A. King stated the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this talk is to prove King's conjecture for the following type of smooth complete toric varieties:    (1) Any d-dimensional smooth toric variety with splitting fan;    (2) Any d-dimensional smooth toric variety with Picard number £ 2;    (3) The blow up of any smooth toric surface at T-invariants points.