Abstract. We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.

We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima's representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on (C 3 ) is irreducible for c ≤ 10.

We construct monads for framed torsion-free sheaves on blow-ups of the complex projective plane at finitely many distinct points. Using these monads we prove that the moduli space of such sheaves is a smooth algebraic variety. Moreover we construct monads for families of such sheaves parameterized by a noetherian scheme S of finite type. A universal monad on the moduli space is introduced and used to prove that the moduli space is fine.

We consider a particular class of holomorphic vector bundles relevant for supersymmetric string theory, called omalous, over nonsingular projective varieties. We use monads to construct examples of such bundles over 3-fold hypersurfaces in P 4 , complete intersection Calabi-Yau manifolds in P k , blow-ups of P 2 at n distinct points, and products P m × P n .

We give an Atiyah–Drinfel’d–Hitchin–Manin (ADHM) description of the Quot scheme of points [Formula: see text] of length [Formula: see text] and rank [Formula: see text] on affine spaces [Formula: see text] which naturally extends both Baranovsky’s representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces [Formula: see text] described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of [Formula: see text] and its punctual version [Formula: see text] where [Formula: see text] is a fixed point on a smooth affine variety [Formula: see text] In this last case, we also study a connectedness result, for some special cases of higher [Formula: see text] and [Formula: see text]