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Question 1. It's not hard to see that $F$ is $\Sigma \times {\Bbb C} P^1$ and $\Sigma$ is a torus. Therefore the Albanese map is a projection to a torus. Question 2. If $f^* \delta=-\delta$, then th …

If the curves are ample (that is, the normal bundles are $\oplus_i\mathcal{O}(k_i)$, with all $k_i>0$), they can be deformed to a smooth one (see Kollar, Rational Curves). Of course, this cannot happe …

You can prove that such threefolds are either P^1-fibration over a base, which is a surface with $h^{2,0}=1$, or have pseudoeffective canonical bundle: arXiv:1304.7891, Corollary 4.3. In the later cas …

There are no holomorphic maps from a projective space to a Kahler manifold X of smaller dimension. Indeed, the pullback of a Kahler form $\omega_X$ cannot be exact, because a positive closed form, in …

Averaging a Kaehler class over the involution, and taking the corresponding Ricci-flat metric, we may assume that the involution preserves a flat metric on a torus. At each fixed point, the eigenvalu …

The group of symplectomorphisms $Aut(X)$ of a K3 is the group $O(\Lambda)$ of automorphisms of its period lattice $\Lambda=H^{1,1}(M,{\Bbb Z})$. For each (-2)-cohomology class $\eta\in H^{1,1}(M,{\B …

There is just no definition of the moduli for complex structures on non-compact manifolds, but by any reasonable definition, it would be (generally) very bad space, certainly infinite-dimensional. For …

This is known, for projective (even Moishezon) manifolds as shown by Dan Popovici in his paper http://arxiv.org/abs/1003.3605 For general Kaehler manifold, this is conjectured. Popovici has proved t …

The second and the third are pretty much equivalent. Indeed, "the period" in XIX century sense is essentially the same as the discrepancy between the branches of a multi-valued function, obtained as a …

If the Chern class vanishes over integers, it's Calabi-Yau manifold, and it has a holomorphic (n,0)-form by Bogomolov's theorem (see there: Two definitions of Calabi-Yau manifolds). If you relax your …

The correct version of this statement is the following. Let $B$ be a stable vector bundle with structure group $G$, and $TB$ a tensor component in the bundle of all tensors over $B$ (that is, a sub-bu …

An obvious obstruction comes from topology: the Chern classes of your bundle should be obtained from restriction of Hodge classes on an ambient variety. This is (more or less) enough to extend a smoo …

It seems that the answer is very simple (and should have been known to Uhlenbeck-Yau). Let $E$ be a stable bundle, equipped with an Hermitian-Einstein metric and connection, and $End(E)$ its authomorp …

This is valid for any vector bundle $B$. One considers $R$ as a 2-form with coefficients in endomorphisms of a bundle $B$. Then $R\wedge R$ is a 4-form with coefficients in endomorphisms, and the trac …

Let me prove that $(L,D)\geq 0$. Consider the set of all (1,1)-classes which satisfy $\eta^2\geq 0$. It is a union of two components $P_+$ and $P_-$, intersecting in 0. If $\eta, \eta'$ are in the …