We discuss some results on categorical crepant resolutions for varieties with quotient singularities. Namely, we prove that under appropriate hypotheses, the derived category of a smooth Deligne-Mumford stack is a strongly crepant non-commutative resolution of singularities of its coarse moduli space.

Abstract. Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space M 0 ort (r, n) of stable rank r orthogonal vector bundles on P 2 , with Chern classes (c 1 , c 2 ) = (0, n), and trivial splitting on the general line, is smooth irreducible of dimension (r − 2)n − r 2 for r = n and n ≥ 4, and r = n − 1 and n ≥ 8. We speculate that the result holds in greater generality.

We define and study the invariance properties of homological units. Some applications are given to the derived invariance of Hodge numbers. In particular, we prove that if X and Y are derived equivalent smooth projective varieties of dimension 4 having the same h 1,1 , then they have the same Hodge numbers. We also give a geometric interpretation of the conjectural invariance of homological units in terms of derived Jacobians.Let X and Y be smooth projective varieties over the field of complex numbers. If X and Y are derived equivalent, there Hochschild homology groups are isomorphic [Orl03]. The Hochschild-Kostant-Rosenberg Theorem [Mar09] then implies the isomorphism of vector spaces:It is still unknown if all the graded pieces which appear in the above decomposition are in fact isomorphic. Namely, the following conjecture is folklore:Conjecture 1.0.1 Let X and Y be smooth projective varieties. Assume that X and Y are derived equivalent, then X and Y have the same Hodge numbers.Note that if X (or Y ) has ample or anti-ample canonical bundle, a famous result of Bondal and Orlov states that derived equivalence of X and Y implies that both varieties are actually isomorphic [BO02]. Hence, conjecture 1.0.1 is interesting when the sign of the canonical bundle of X (and Y ) is not definite. For instance if X and Y have trivial canonical bundle.In dimension 1 and 2 (and also in dimension 3, if X and Y have trivial canonical bundle), this conjecture is an obvious consequence of the derived invariance of Hochschild homology, the Hochschild-Kostant-Rosenberg isomorphism and the Hodge symmetry. In dimension 3, without any assumption on the canonical bundle, the conjecture is still true and has been settled by Popa and Schnell [PS11]. If one assumes that the derived equivalence comes from a birational correspondence then the conjecture has been proved by Orlov [Orl05]. In dimension 4 or higher (and without the birational assumption), nothing is known, even if the canonical bundles of X and Y are trivial.Leaving aside this hard problem, one could be interested in a slightly less ambitious question: Question 1.0.2 Let X and Y be two smooth projective varieties. Assume that X and Y are derived equivalent. Denote by HH • (X) (resp. HH • (X)), the Hochschild homology graded vector space (resp. Hochschild cohomology graded algebra) of X.• Are there non-trivial, canonically defined, sub-vector spaces of HH • (X) and HH • (Y ) which are isomorphic?• Are there non-trivial, canonically defined, sub-algebras of HH • (X) and HH • (Y ) which are isomorphic?1. The Fourier-Mukai kernel representing Φ is a (possibly shifted) generically pure vector bundle 1 on X × Y ,

Let X ⊂ P N be an irreducible, non degenerate projective variety and let X * ⊂ P N * be its projective dual. Let L ⊂ P N be a linear space such that L, T X,x = P N for all x ∈ X smooth and such that the lines in X meeting L do not cover X. If x ∈ X is general, we prove that the multiplicity of X * at a general point of L, T X,x ⊥ is strictly greater than the multiplicity of X * at a general point of L ⊥ . This is a strong refinement of Bertini's theorem.