International Congress of Mathematicians

54 A.Bondal D.OrlovIn particular, we expect that T> b (coh(£nd(q*ÖY))) has a fully faithful functorinto V b (X) for any (commutative) resolution of X. Moreover, if the resolution iscrêpant then the functor has to be an equivalence.Let X be the quotient of a smooth Y by an action of a finite group G. Ifthe locus of the points in Y with nontrivial stabilizer in G has codimension > 2,then the category of coherent £nd(g»öy )-modules is equivalent to the category ofG-equivariant coherent sheaves on Y. Therefore the conjecture is a generalizationof the derived McKay correspondence due to Bridgeland-King-Reid [BKR].6. Complete intersection of quadrics and noncommutativegeometryThis section is related to the previous one by Grothendieck slogan that projectivegeometry is a part of theory of singularities.Let X be a smooth intersection of two projective quadrics of even dimensiond over an algebraically closed field of characteristic zero. It appears that if weconsider the hyperelliptic curve C which is the double cover of P 1 that parameterizesthe pencil of quadrics, with ramification in the points corresponding to degeneratequadrics, then V b (C) is embedded in V b (X) as a full subcategory [BOI]. This givesa categorical explanation for the classical description of moduli spaces of semistablebundles on the curve C as moduli spaces of (complexes of) coherent sheaves on X.The orthogonal to V b (C) in V b (X) is decomposed into an exceptional sequence(of line bundles ). More precisely, we have a semiorthogonal decompositionV b (X) = (öxhd + 3),...,öx,'D b (C)). (6.1)When a greater number of quadrics is intersected, objects of noncommutativegeometry naturally show up: instead of coherent sheaves on hyperelliptic curveswe must consider modules over a sheaf of noncommutative algebras. More aboutnoncommutative geometry is in the talk of T. Stafford at this Congress.Consider a system of m quadrics in V(V), i.e. a linear embedding U ^y S 2 V*,where dimU = m, dimV = n, 2m < n. Let X, the complete intersection ofthe quadrics, be a smooth subvariety in W(V) of dimension n — m — 1. Let A =® H 0 (X,O(ij) be the coordinate ring of X. This graded quadratic algebra isKoszul due to Tate [Ta]. The quadratic dual algebra B = A is the generalizedhomogeneous Clifford algebra. It is generated in degree 1 by the space V, therelations being given by the kernel of the dual to map S 2 V —¥ U*, viewed as asubspace in V V. The center of B is generated by U* (a subspace of quadraticelements in B) and an element d, which satisfies the equation d 2 = f where / isthe equation of the locus of degenerate quadrics in U. Algebra B is finite over thecentral subalgebra S = S'U*. The Veronese subalgebra B ev = (BB 2 ì is finite overthe Veronese subalgebra S ev = ®S 2% U*. Since Proj S ev is isomorphic to V(U),the sheafification of B ev over Proj S ev is a sheaf B of finite algebras over ö V ( V yConsider the derived category T> b (coh(Bj) of coherent right B-modules.