# Moishezon manifolds with vanishing first Chern class

Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?

This is true whenever $M$ is Kähler (and therefore projective) and was proved independently by Bogomolov, Fujiki and Lieberman. It is also a well-known consequence of Yau's solution of the Calabi Conjecture. Also when $\mathrm{dim}M=2$ then $M$ is automatically projective, so the question is really about dimensions $3$ or more.

The only examples that I know of non-Kähler Moishezon manifolds with $c_1(M)=0$ are obtained by applying a Mukai flop to a projective hyperkähler manifold, and so they have holomorphically trivial canonical bundle. They are described here.

Are there other simple examples of such manifolds?

The same question can also be asked for compact complex manifolds bimeromorphic to Kähler (i.e. in Fujiki's class $\mathcal{C}$).

• Oguiso, Keiji:Two remarks on Calabi-Yau Moishezon threefolds. J. Reine Angew. Math. 452 (1994), 153–161.
– user21574
Jul 22 '17 at 17:18

Take a threefold with $n$ ordinary double points and trivial canonical divisor. Then it has $2^n$ small resolutions of singularities. Each of those is a Moishezon manifold (typically non-projective) with $c_1 = 0$.

• Thanks for these examples, which indeed have trivial canonical bundle. Do you have any ideas about my main question? Jul 29 '12 at 22:40
• I am not an expert in non-Kahler varieties, so maybe I just don't understand where the problem lies. But it seems to me that since an appropriate blouwup of $M$ is projective it should follow that $K_M$ is defined as an element of $H^2(M,{\mathbf Z})$ and so its vanishing after tensoring with ${\mathbf R}$ should imply that it is torsion. Jul 30 '12 at 7:18
• Yes, it implies that its first Chern class is torsion in $H^2(M,\mathbb{Z})$, but this is far from saying that $K_M$ is torsion in $\mathrm{Pic}(M)$, i.e. that some positive power of it has a never-vanishing global holomorphic section. See also the related question mathoverflow.net/questions/15003/… (which is about Kahler manifolds) Jul 30 '12 at 13:44

Added. I just realised that the statement concerning Moishenzon manifolds holds in dimension up to $4$. In dimension three this is a corollary of minimal model programme and in dimension $4$ this follows from Theorem 0.4 here:

http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/coneduality.pdf

In dimensions higher than four the statement would follow from the main conjecture of minimal model programm it states that a projective manifold with pseudoeffective canonical class has non-negative Kodaira dimension. This conjecture holds if dimension less than $4$ and for dimension four the above article can be used instead of it for our purpose.

Indeed, every Moishenson manifold admits a blow up that is projective. According to the condition that you state the canonical class of the blow up is pseudoeffective. So according to the conjecture a power of the canonical bundle on the blow up has a non-zero section. Such a section should vanish on the exceptional divisors of the blow up. So, I guess you should be able to push it down to the original manifold (again to a section of the power of the canonical bundle). This section would not vanish since it could vanish only on a hypersurface and this would mean the $c_1\ne 0$ (since there are plenty of curves on Moishenson manifolds).

More details. The canonical class of the blow up is pseudoeffective beacuse it is positive on every coverening family of curves. In particular the blow up is not unirulled. So it should have non-negative Kodaira dimension according to conjecture 1.6 here (page 6): https://www.dpmms.cam.ac.uk/~cb496/birgeom-paris-public.pdf

• Are you sure that the canonical bundle of the blowup is nef? How about if $M$ is a surface with $K_M=0$ and I blow up a point? Then the canonical bundle of the blowup is linearly equivalent to the exceptional divisor $E$, which is not nef. Dec 16 '12 at 5:44
• YangMills, I meant not nef, but pseudoeffective, of course, thanks for correcting me :) I rewrote the answer correspondingly. Dec 16 '12 at 9:30

This problem is solved affirmatively in Theorem 1.5 in this paper.

The idea is that after some blowups we obtain a compact Kähler manifold whose canonical bundle is effective after twisting by a numerically trivial line bundle. A seminal result of Simpson, recently extended to the Kähler case by Wang, then shows that the manifold has nonnegative Kodaira dimension (this is related to the proof of the abundance conjecture when the numerical dimension of the canonical bundle is zero).

Once we know that there is a nontrivial pluricanonical section, it is easy to conclude that this section must be never vanishing.