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A hypercomplex manifold $M$ is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it is locally obtained as a Hessian of a function averaged with quaternions. HKT metric is a natural analogue of a Kahler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of Buchdahl-Lamari's theorem for complex surfaces. Buchdahl and Lamari have shown that a complex surface M admits a Kahler structure iff $b_1(M)$ is even. We show that a hypercomplex manifold M with Obata holonomy $SL(2,{\mathbb H})$ admits an HKT structure iff $H^{0,1}(M)=H^1({\cal O}_M)$ is even., 30 pages. arXiv admin note: text overlap with arXiv:0808.3218, arXiv:1009.1178

In [2] , Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori T R 2 n for any n ≥ 3 . We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.

In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda., Comment: 14 pages, 0 figures. Revised version

In this paper we prove a compactness result for compact Kahler Ricci gradient shrinking solitons. If ( M i , g i ) is a sequence of Kahler Ricci solitons of real dimension n ⩾ 4 , whose curvatures have uniformly bounded L n / 2 norms, whose Ricci curvatures are uniformly bounded from below and μ ( g i , 1 / 2 ) ⩾ A (where μ is Perelman's functional), there is a subsequence ( M i , g i ) converging to a compact orbifold ( M ∞ , g ∞ ) with finitely many isolated singularities, where g ∞ is a Kahler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kahler Ricci soliton equation in a lifting around singular points).

We define and study a new numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^{n-1}., Comment: Significantly strenghtened, compared to the earlier version. To appear in Advances in Mathematics, 41 pages

In this paper we show the existence of stable symplectic non-holomorphic two-spheres in Kahler manifolds of positive constant scalar curvature of real dimension four and in Kahler–Einstein Fano manifolds of real dimension six. Some of the techniques used involve deformation theory of algebraic cycles.

Atiyah and Hirzebruch gave examples ofeven degree torsion classes in the singularcohomology of a smooth complex projective manifold, which arenot Poincar\'{e} dual to an algebraiccycle. We notice that the order ofthese classes are small comparedto the dimension of the manifold.However, building upon a construction ofKoll\`{a}r, one can provide such examples witharbitrary high prime order, the dimension being fixed. This method alsoprovides examples of torsion algebraiccycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration onChow groups., Comment: Final version, to appear in Adv. in Mathematics, special issue dedicated to M. Artin. References to related work by C. Schoen added. A mistake mentioned by K. O'Grady corrected