1. There are curves called stable curves of Deligne-Mumford. We know first smooth curves, then comes stable curves, relatively simple curves with mild singularities and finite automorphism. 2. The moduli Mg of stable curves of genus g, it is now known to be a projective scheme. 3. The moduli Mg of smooth curves of genus g is not compact, only quasi-projective. It is compactified as the moduli M...

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Goren...

Two types of tests are available for studying personality. These are the paper-pencil tests and the projective tests. Projective techniques which reveal the subject's personality through his treatment of the test material are generally more valid and more revealing than the paper-pencil tests which require the underlining of words or the answering of "Yes" or "No" to a list of questions. This i...

During the last years there has been growing interest in vector bundles with additional structures, e.g. parabolic and level structures. This paper results from an attempt to construct quasi-projective moduli spaces for framed bundles, i.e. bundles together with an isomorphism to a fixed bundle on a divisor as introduced in [Do], [L1] and [Lü]. More generally one can ask for bundles with a homo...

We describe some of the connections between the Bieri–Neumann– Strebel–Renz invariants, the Dwyer–Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, tran...

In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov’s theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees > 1) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi–projective variety with an irreducible smooth divisor at infinity. We defi...