We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.

We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety, we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application, we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the literature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

Let X be an irreducible holomorphic symplectic manifold of K3 [n] deformation type, n ≥ 2. There exists over X ×X a rank 2n−2 rigid and stable reflexive sheaf A of Azumaya algebras, constructed in [Ma5], such that the pair (X × X, A) is deformation equivalent to the following pair (M × M, AM ). The manifold M is a smooth and compact moduli space of stable sheaves over a K3 surface S, and AM is the reflexive sheaf, whose fiber, over a pair (F1, F2) ∈ M × M of non-isomorphic stable sheaves F1 and F2, is End Ext 1 S (F1, F2) . We prove in this paper the following uniqueness result.Let A1 and A2 be two rigid Azumaya algebras over X × X, which are (ω, ω)-slope-stable with respect to the same Kähler class ω on X, and such that (X × X, Ai) is deformation equivalent to (M × M, AM ), for i = 1, 2. Assume that the singularities of A1 and A2 along the diagonal are of the same infinitesimal type prescribed by AM . Then A1 is isomorphic to A2 or A * 2 . Furthermore, if P ic(X) is trivial, or cyclic generated by a class of non-negative Beauville-Bogomolov-Fujiki degree, then there exists a unique pair {A, A * }, such that (X × X, A) is deformation equivalent to (M × M, AM ), and A is (ω, ω)-slope-stable with respect to every Kähler class ω on X.The above is the main example of a more general global Torelli theorem proven. The result is used in the authors forthcoming work on generalized deformations of K3 surfaces. EYAL MARKMAN AND SUKHENDU MEHROTRA 4.2. Local moduli space of reflexive Azumaya algebras 31 4.3. A global moduli space 34 5. Stability and separability 35 6. A global Torelli theorem 36 7. Proof of Theorem 1.11 establishing the main example 38 References 43From now on the term a sheaf of Azumaya algebras will mean a reflexive sheaf of Azumaya O X -algebras. A subsheaf P of a sheaf E of Azumaya algebras is a sheaf of maximal parabolic subalgebras if, in the notation of Definition 1.1, η α (P | Uα ) is the sheaf of subalgebras of End(F α )1 Caution: The standard definition of a sheaf of Azumaya OX -algebras assumes that E is a locally free OXmodule, while we assume only that it is reflexive.

We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.

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Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M × M there exists a rank 2n − 2 reflexive hyperholomorphic sheaf EM , whose fiber over a non-diagonal point (F1, F2) is Ext 1 S (F1, F2). The sheaf EM can be deformed along some twistor path to a sheaf EX over the cartesian square X × X of every Kähler manifold X deformation equivalent to M . We prove that EX is infinitesimally rigid, and the isomorphism class of the Azumaya algebra E nd(EX ) is independent of the twistor path chosen. This verifies conjectures in [MM1, MM2] and renders the results of these two papers unconditional.

Abstract. Given a compact connected Riemann surface X of genus g ≥ 2, and integers r ≥ 2, d p > 0 and d z > 0, in [BDHW], a generalized quot scheme Q X (r, d p , d z ) was introduced. Our aim here is to compute the holomorphic automorphism group of Q X (r, d p , d z ). It is shown that the connected component of Aut(Q X (r, d p , d z )) containing the identity automorphism is PGL(r, C). As an application of it, we prove that if the generalized quot schemes of two Riemann surfaces are holomorphically isomorphic, then the two Riemann surfaces themselves are isomorphic.