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There are many such examples, for instance, all complex nilmanifolds (and most complex solvmanifolds) have tangent bundle which is topologically trivial. The Hopf manifolds also have topologically tri …

There is no such thing: any $\phi$-invariant exact 1-form is a differential of $\phi$-invariant function. Indeed, let $\alpha$ be an exact $\phi$-invariant form, $\alpha=df$, where $f$ is not $\phi$-i …

closed (1,1)-forms and currents on X are not necessary locally $dd^c$-exact in general What makes it different when X is singular? The obstruction to local $dd^c$-lemma is $R^1\pi_*(O_{X'})$, where …

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))= …

Let $(M,g)$ be a Riemannian manifold. The Riemannian cone of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$. A manifold is called Sasakian if its cone is Kähler, with t …

Sure, any closed 2-form $\eta$ with integer cohomology can serve as the curvature of a connection on a line bundle. This can be seen if you take a line bundle with the same Chern class and connection …

I suppose the following result follows from Ambrose-Singer theorem, but I cannot find a reference, and the arguments I found in the literature are usually weaker. The idea is that holonomy over a null …

This observation seems to be very easy, but it takes care of many examples. Suppose that $A$ corresponds to a contraction (that is, all eigenvalues are $< 1$ in absolute value). Decompose ${\Bbb R}^n …

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 …

Question. Is there a holomorphic tubular neighbourhood of S in M? I have a counterexample. Let $\phi:\; Z^2 \to Aut(C^n)$ be a group homomorphism, with $Aut(B)$ denote the group of holomorphic autom …

To put it differently, is it true that X locally looks like a cotangent bundle? This is false. Indeed, take an elliptic curve $C$ inside an elliptic K3 surface. If it had a neighbourhood $U$ which is …

For $SU(3)$ and Hopf it is the anticanonical bundle which has many sections. In other examples of Joyce, it is also the anticanonical bundle or its powers. For nilmanifolds, the canonical bundle is ho …

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 …

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 …

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 …