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A paper by Gabor Etesi was published that purports to solve a major outstanding problem: Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, 04 …

I have looked for a while for a proof which does not use the Calabi-Yau theorem and nobody seems to know it. Also, there are plenty of non-Kaehler manifolds with canonical bundle trivial topologicall …

In complex dimension 3 or more it is still an open conjecture (which was re-stated Yau a couple of years ago in his UCLA lectures). There is not a single known example of an almost complex manifold of …

The main obstruction to existence of Kahler metric (in addition to Lefschetz SL(2)-action and Riemann-Hodge relations in cohomology) is homotopy formality: the cohomology ring of a Kahler manifold is …

Since Grassmannian $Gr(n,m)=SO(n+m)/SO(n)\times SO(m)$ is a homogeneous manifold, you can take any Riemannian metric, and average with $SO(n+m)$-action. Then you show that an $SO(n+m)$-invariant metri …

There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold, is also deformationally equivalent to a hyperkaehler manif …

There is a well known problem of LeBrun-Salamon: are there any non-symmetric compact quaternionic-Kahler manifolds of positive scalar (and Ricci) curvature? It is hard and still unsolved: Quaternionic …

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to …

The most general version of Weitzenbock identities (with coefficients in appropriate universal enveloping algebras) is due to Uwe Semmelmann and Gregor Weingart: http://arxiv.org/abs/math/0702031 "The …

What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions (locall …

In his thesis, Bieberbach solved Hilbert 18 problem and proved that any compact, flat Riemannian manifold is a quotient of a torus. I need a reference to an orbifold version of this result: any comp …

The corresponding Kahler metric is called "Weil-Petersson metric"; it is often constructed using infinite-dimensional determinants of the corresponding Laplace operators. The standard reference is a s …

Let $\alpha$ be an exterior product of a harmonic and a parallel form on a Riemannian manifold. Then $\alpha$ is known to be harmonic. I have heard that this is an old result due to R. Bott, but I cou …

The Hodge * operator action on cohomology is generally speaking metric-dependent, hence * is not well-defined without fixing the metric. There are some caveats. On complex curves, for example, the Hod …

It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))= …