In the paper [MTT] a conceptuel description of compactifications of moduli spaces of stable vector bundles on surfaces has been given, whose boundaries consist of vector bundles on trees of sufaces. In this article a typical basic case for the projective plane is described explicitly including the constrution of a relevant Kirwan blow up.

We study relatively semi-stable vector bundles and their moduli on non-Kähler principal elliptic bundles over compact complex manifolds of arbitrary dimension. The main technical tools used are the twisted Fourier-Mukai transform and a spectral cover construction. For the important example of such principal bundles, the numerical invariants of a 3-dimensional non-Kähler elliptic principal bundle over a primary Kodaira surface are computed.

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c1 = 0, c1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers. MSC: 14J60, 14D20, 14D21

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.

We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties.

We study relatively semi-stable vector bundles on non-K\" ahler principal elliptic bundles, which are Calabi-Yau type 3-folds. The main technical tools used are twisted Fourier-Mukai transform and spectral cover construction.

We consider the moduli space of stable vector bundles on curves embedded in P_2 with Hilbert polynomial 3m+1 and construct a compactification of this space by vector bundles. The result is a blow up of the Simpson moduli space M_{3m+1}(P_2).

We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.

We show that the idea used by Kempf (1990) in order to obtain a splitting criterion for vector bundles on projective spaces leads to an elementary proof of the Babylonian tower theorem for this class of bundles, a result due to Barth--Van de Ven (1974) in the rank 2 case and to Sato (1977) and Tyurin (1976) in the case of arbitrary rank. As a byproduct we obtain a slight improvement of the numerical criterion of Flenner (1985) in the particular case under consideration.

We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, essentially translating into more fancy terms the results of Trautmann (1978) which, in turn, were influenced by ideas of Horrocks (1964), (1980). However, the result expressed by Theorem 5.1 and its corollary, which establishes a relation between the Tate resolutions over the exterior algebra ( described in a paper by Eisenbud, Floystad, and Schreyer) and HT-complexes, might be new, although, perhaps, not a surprise to experts.

We prove that the space of mathematical instantons with second Chern class 5 over ℙ_3 is smooth and irreducible. Unified and simple proofs for the same statements in case of second Chern class ≤ 4 are contained.

Mathematical instanton bundles on P_3 have their analogues in rank–2n instanton bundles on odd dimensional projective spaces P_2n+1. The families of special instanton bundles on these spaces generalize the special 'tHooft bundles on P_3. We prove that for a special symplectic instanton bundle E on P_2n+1 with c_2=k h^1End( E) = 4(3n-1) k + (2n-5)(2n-1). Therefore the dimension of the moduli space of instanton bundles grows linearly in k. The main difference with the well known case of P_3 is that h^2End( E) is nonzero, in fact we prove that it grows quadratically in k. Special symplectic instanton bundles turn out to be singular points of the moduli space. Such bundles E are SL(2)–invariant and the result is obtained regarding the cohomology groups of E as SL(2)–representations.

Dedicated to the memory of Constantin Banica Contents Introduction 1 1 Multiple extensions and Koszul structures on lines 3 2 The Beilinson resolution 7 3 Instanton bundles with one linear section 10 4 Extensions of Koszul structures and elementary transformations 14 5 Normal forms of monad arrows 15 References 16 Introduction In this paper we prove that the moduli scheme MI(n) of mathematical instanton bundles over IP 3 with second Chern class n is smooth at bundles E with h 0 E(1) 6 = 0. By a result in 6] this number is 2. In case h 0 E(1) = 2 of special 't Hooft bundles the smoothness is a result of A. Hirschowitz-M.S. Narasimhan in 8]. This case is included in our proof. In the remaining general situation h 0 E(1) = 1, the reduced zero scheme Z red of the unique section is a disjoint union of lines, but the scheme Z itself has nilpotent structure in general. Here we cannot deform the scheme Z as a zero scheme. We study these nilpotent structures, which turn out to have resolutions 0 ! nO(?2) ! 2nO(?1) ! nO ! O Z ! 0 given by nice regular matrices, and which are self-dual, see 1.1. We call such structures of Koszul type or Koszul structures. They are exactly the primitive structures of type O ` in the notation of C. Banica-O. Forster, 5], see Corollary 1.9. In proposition 2.3 we show that a multiple structure X on a line, which admits a resolution of the above type with unspeciied matrices, and which satisses ! X = O X (?2), is already a Koszul structure given by special matrices as in 1.2. It follows from this that the zero scheme Z of the unique section of E(1) as above is a disjoint union of Koszul structure. This enables us to prove H 1 N _ Z (1) = 0 for the conormal sheaf of Z, which in turn implies Ext 2 (E; E(?1)) = 0 and Ext 2 (E; E) = 0, and hence smoothness of MI(n) at E. The Koszul structures had occured already implicitly in 10] in the description of elementary transformations of instanton bundles along lines. It is remarkable, see section 4, that extensions of Koszul structures are closely related to these elementary transforms. The paper is closed by presenting a normal form for the right part of …

Let M(0, 2) denote the moduli space of rank 2 vector bundles on P3, with cl = 0, c2 = 2. We denote by M(0, 2) the schematic closure of M(0, 2) in the Maruyama scheme of rank 2 semi-stable sheaves ön P3 with c± = 0, c2 — 2, c3 = 0 and by M(0,2) the normalisation of M(0, 2). In [NT] we had constructed a normal projective variety Q which is a quadric bündle ("Poncelet bündle") over the Hubert scheme C(G) of conics in the Grassmannian of projective lines in P3 and a birational surjective morphism Q -> jßf(0, 2) such that the exceptional set is of codimension ^ 2 (in fact of codimension 5). From this it follows that Cl(0 = C1M (0,2) (where 'CP denotes the set of classes of Weil divisors on a normal variety) and that M(0, 2) is not locally factorial.

We consider in this lecture the degenerations of “physical instantons” as algebraic vector bundles on ℙ3(ℂ). These are the vector bundles which correspond to self-dual SU(2)— Yang-Mills potentials on the 4-sphere by the Atiyah-Ward correspondence [l]. In 2. we characterise most of the sheaves in ℙ3(ℂ) which are degenerations of instantons. In 3. we give an algebraic construction of the Donaldson compactification of I(n), and in 4. we interprete the degenerations as global equivalence classes of singular connections in the generic case, thereby obtaining a precise algebraic description of bubbling out of instantons. Details can be found in [3] and [4].