Remarks on astheno-Kahler manifolds, Bott-Chern and Aeppli cohomology groups
We provide a new cohomological obstruction to the existence of astheno-Kahler metrics, and study relevant examples.
math.DG↗
arXiv subjects
Rares Rasdeaconu
Publications and source records attributed to Rares Rasdeaconu.
On the classification of ALE Kähler manifolds
The underlying complex structure of an ALE Kähler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE Kähler surfaces with a given group at infinity.
math.DG↗
Balanced metrics on uniruled manifolds
We show that an $n-$dimensional Moishezon manifold is uniruled if and only if it supports a balanced metric $ω^{n-1}$ of positive total scalar Chern curvature. A similar statement also holds true for class $\mathcal C$ manifolds of dimension three.
math.DG↗
Balanced Manifolds and SKT Metrics
The equality between the balanced and the Gauduchon cones is discussed in several situations. In particular, it is shown that equality does not hold on many twistor spaces, and it holds on Moishezon manifolds. Moreover, it is proved that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon cone is in fact Kahler.
math.CV↗
Qualitative aspects of counting real rational curves on real K3 surfaces
We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic scale, the rate of growth of the number of such real rational curves is, up to a constant factor, the rate of growth of the number of complex rational curves. We indicate a few instances when the lower bound for the number of real rational curves provided by our count is sharp. In addition, we exhibit various congruences between real and complex counts.
math.AG↗
ALE Ricci-flat Kahler surfaces and weighted projective spaces
We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities as log del Pezzo surfaces with only cyclic quotient singularities at infinity.
math.DG↗
Counting real rational curves on K3 surfaces
We provide a real analog of the Yau-Zaslow formula counting rational curves on $K3$ surfaces.
math.AG↗
On Normalized Ricci Flow and Smooth Structures on Four-Manifolds with $b^+=1$
We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds ${\mathbb C}{P}^2 # l \overline{{\mathbb C}{P}^2}$, where $5 \leq l \leq 8$.
math.DG↗
Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$
We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.
math.DG↗
The Algebraic Rational Blow-Down
The normal connected sum construction of Gompf and the rational blowing-down technique of Fintushel - Stern are important tools in constructing symplectic 4-manifolds. In some cases, the 4-manifolds created this way are of Kahler type. In this article we investigate the occurrence of this phenomenon and give relevant examples.
math.SG↗
The Kodaira dimension of diffeomorphic Kähler 3-folds
We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic K\" ahler threefolds.
math.DG↗