Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity An. We first study the analogue for the other Kleinian singularities. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau 2-dimensional category.

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of A n -singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for A n -singularities.

It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain non-commutative algebra. In this paper, we present a method to construct a tilting generator from an ample line bundle, and construct it in several examples.

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf O X generates the derived category D b (X) for smooth projective toric varieties X.Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this article, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.

Let X and Y be smooth projective varieties over C . We say that X and Y are D-equivalent (or, X is a Fourier-Mukai partner of Y ) if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories. The aim of this short note is to find an example of mutually D-equivalent but not isomorphic relatively minimal elliptic surfaces.