Let V be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of V . We show that the m-canonical map of V is birational for all m≥73 and that the canonical volume Vol(V ) ≥ 1/2660 . When (OV ) ≤1, our result is Vol(V )≥1/420 , which is optimal. Other effective results are also included in the paper.

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We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

Let H be a semisimple algebraic group. We prove the semistable reduction theorem for μ–semistable principal H–bundles over a smooth projective variety X defined over the field C. When X is a smooth projective surface and H is simple, we construct the algebro–geometric Donaldson–Uhlenbeck compactification of the moduli space of μ–semistable principal H–bundles with fixed char- acteristic classes and describe its points. For large characteristic classes we show that the moduli space of μ–stable principal H–bundles is non–empty.

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology.