Related papers: Classification of Poor Manifolds in Low dimensions

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold…

Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, until now the only known examples with the maximal possible rank four are the real Kaehler minimal…

We prove that every compact K\"ahler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal…

For an arbitrary submanifold $M \subset \mathbb{C}P^n$ we determine conditions under which it is austere, i.e., the normal bundle of $M$ is special Lagrangian with respect to Stenzel's Ricci-flat K\"ahler metric on $T\mathbb{C}P^n$. We also…

We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they a fortiori are not deformation equivalent to a projective…

Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in…

We give an introduction to the compactification of the moduli space of surfaces of general type introduced by Koll\'ar and Shepherd-Barron and generalized to the case of surfaces with a divisor by Alexeev. The construction is an application…

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in…

We give four constructions of non-$\partial\bar\partial$ (hence non-K\"ahler) manifolds: (1) A simply connected page-$1$-$\partial\bar\partial$-manifold (2) A simply connected $dd^c+3$-manifold (3) For any $r\geq 2$, a simply connected…

Using results of our papers [19], [20] and [21] about classification of degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, we classify Picard lattices of Kahlerian K3 surfaces. By classification we understand…

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular…

For every compact K\"ahler manifold $X$ of algebraic dimension $a(X) = \dim X - 1$, we prove that $X$ has arbitrarily small deformations to some projective manifolds.

We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy…

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

The K\"ahler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on K\"ahler manifolds as a measure of ``k\"alerianity''. Here we give a partial classification of compact complex surfaces of K\"ahler…

In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the…

Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case…

Let $X$ be a smooth complex projective variety of dimension three and let $L$ be an ample line bundle on $X$. In this paper, we provide a lower bound of the dimension of the global sections of $m(K_{X}+L)$ under the assumption that…

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a…