International Congress of Mathematicians

Derived Categories of Coherent Sheaves 51The criterion is a particular manifestation of the following important principle:suppose M is realized as an appropriate moduli space of pairwise homologicallyorthogonal objects in a triangulated category V taken 'from real life', then one canexpect a sheaf of finite (noncommutative) algebras AM over OM and a fully faithfulfunctor from the derived category V b (coh(AM)) of coherent modules over AM to V.There are also strong indications that this principle should have a generalization,at the price of considering noncommutative DG moduli spaces, to the casewhen the orthogonality condition is dropped.4. Derived categories and birational geometryBehavior of derived categories under birational transformations shows thatthey can serve as a useful tool in comprehending various phenomena of birationalgeometry and play possibly the key role in realizing the minimal model program.First, we give a description of the derived category of the blow-up of a varietywith smooth center in terms of the categories of the variety and of the center. Let Ybe a smooth subvariety of codimension r in a smooth algebraic variety X. Denote Xthe smooth algebraic variety obtained by blowing up X along the center Y. Thereexists a fibred square:Y AXp 4- 7T 4-Y A xwhere i and j are smooth embeddings, and p:Y—¥ Y is the projective fibration ofthe exceptional divisor Y in X over the center Y. Recall that Y = W(N X / Y ) is theprojective normal bundle. Denote by 0 Y (1) the relative Grothendieck sheaf.Proposition 7 [Ori] Let L be any invertible sheaf on Y. The functorsLTT* :V b (X)—yV b (X),are fully faithful.Hj4L®p*(-j): V b (Y) —• V b (X)Denote by D(X) the full subcategory of V b (X) which is the image of V b (X)with respect to the functor hn* and by D(Y)k the full subcategories of V b (X) whichare the images of V b (Y) with respect to the functors Rj»(öy(fc) ®p*(-)).Theorem 8 [Orl][B01] We have the semiorthogonal decomposition of the categoryof the blow-up:V b (X) = (D(Y)^r+ i,...,D(Y)^i,D(X)).Now we consider the behavior of the derived categories of coherent sheaveswith respect to the special birational transformations called flips and flops.