Introduction to lie algebras and representation theory

Let X and Y be smooth projective varieties over ℂ. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, and K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.

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D-Equivalence and K-Equivalence

Strong rigidity of positive quaternion Kähler manifolds.

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds and show that has the structure of a commutative associative topological monoid. The set of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in is not closed under this multiplication; however, the join of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

On sasakian–einstein geometry

A smooth variety X over complex numbers is called Fano if its anticanonical divisor — Kx is ample. The purpose of this note is to collect some results on contractions of extremal rays (Mori theory) which are interesting from the point of describing Fano manifolds of dimension ^ 4. The theory has been successfully used in dimension 3, see [MM], and there is some hope that a similar approach may work somehow in higher dimensions. Furthermore, contractions of extremal rays of Fano manifolds can be treated äs testing grounds for contractions of rays in general. Although contractions of rays of smooth varieties are well understood in dimensions 2 and 3, see [M2], there are still few things known in higher dimensions, see [An], [Bei] and [Ka]. Most of the results of sections l and 2 of the present paper are applicable in a general set-up, i.e. do not need the assumption on ampleness of

Jarosław A. Wiśniewski

Papers

On a conjecture of Mukai

On manifolds whose tangent bundle contains an ample subbundle

Length of extremal rays and generalized adjunction

On fano manifolds of large index

A note on nonvanishing and applications