Hi everyone! Peter, Allison, and Laura have graciously allowed me to join them as an author.
I thought I would start by hashing out the relation between monotone and Fano. It is not a subtle relationship, but somehow I end up working it out for myself every six months. The idea is that they are basically the same for projective Kaehler manifolds:
Theorem. If is a projective monotone Kaehler manifold, then it is Fano. If is a Fano complex manifold, then there exists a symplectic form on so that is a monotone Kaehler manifold.
Recall that a symplectic manifold is monotone if for , where the first Chern class is defined using an almost-complex structure compatible with . A complex manifold is Fano if there is an immersion and a positive integer so that , where is the dual of the canonical bundle.
Let’s start with the Fano-implies-monotone direction. Fix and as above. If is the Fubini-Study form on , then . Define ; I claim that is monotone. The compatibility of with is obvious. The first Chern class of a line bundle equals the first Chern class of its top exterior power more-or-less by definition, so: . But is a symplectic form? It is closed since is closed. The nondegeneracy of follows from the fact that if is a Kaehler manifold and is a complex submanifold, then is nondegenerate. This is easy: given , the Kaehler condition gives .
The monotone-implies-Fano direction is less elementary. It relies on the Nakai-Moishezon-Kleiman criterion (see Lazarsfeld’s first positivity book):
Theorem. Let be a line bundle on a projective scheme . Then is ample if and only if for every positive-dimensional irreducible subvariety .
So, let be a projective monotone Kaehler manifold. To show that is Fano it is enough to show that for every positive-dimensional complex submanifold , . But , so this integral is equal to , which is positive.
Things are this straightforward only in the projective Kaehler setting. E.g. there are various generalizations of “Fano” to the symplectic setting. McDuff–Salamon say that an almost-Kaehler manifold is symplectic Fano if for every nonzero that can be represented by a -curve. Clearly monotone implies symplectic Fano, but the converse is false: e.g. there are 2-dimensional complex tori that have no holomorphic curves (hence are symplectic Fano) but are not projective (hence are not Fano) and do not admit an -compatible monotone symplectic form (thanks John!).