The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded derived categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective, then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.
Abstract. Each choice of a Kähler class on a compact complex manifold defines an action of the Lie algebra sl(2) on its total complex cohomology. If a nonempty set of such Kähler classes is given, then we prove that the corresponding sl(2)-copies generate a semisimple Lie algebra. We investigate the formal properties of the resulting representation and we work things out explicitly in the case of complex tori, hyperkähler manifolds and flag varieties. We pay special attention to the cases where this leads to a Jordan algebra structure or a graded Frobenius algebra.
We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.
We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.
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