Monthly Archives: March 2013

Monotone vs. Fano

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 (X, \omega, J) is a projective monotone Kaehler manifold, then it is Fano.  If (X, J) is a Fano complex manifold, then there exists a symplectic form \omega on X so that (X, \omega, J) is a monotone Kaehler manifold.

Recall that a symplectic manifold (X, \omega) is monotone if [\omega] = \lambda c_1(TX) for \lambda > 0, where the first Chern class is defined using an almost-complex structure compatible with \omega.  A complex manifold (X^n, J) is Fano if there is an immersion \varphi: X \hookrightarrow \mathbb{CP}^N and a positive integer k so that (K_X^{-1})^{\otimes k} = \varphi^*\mathcal{O}(1), where K_X^{-1} = \Lambda^{n,0}TX is the dual of the canonical bundle.

Let’s start with the Fano-implies-monotone direction.  Fix (X^n, J) and \varphi: X \hookrightarrow \mathbb{CP}^N as above.  If \omega_{\text{FS}} is the Fubini-Study form on \mathbb{CP}^N, then c_1(T\mathbb{CP}^N) = [\omega_{\text{FS}}].  Define \omega_X := \varphi^*\omega_{\text{FS}}; I claim that (X, \omega_X) is monotone.  The compatibility of \omega_X with J 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: k c_1(TX) = k c_1(K_X^{-1}) = c_1((K_X^{-1})^{\otimes k}) = c_1(\varphi^*O(1)) = [\omega_X].  But is \omega_X a symplectic form?  It is closed since \omega_{\text{FS}} is closed.  The nondegeneracy of \omega_X follows from the fact that if Y is a Kaehler manifold and Z \subset Y is a complex submanifold, then \omega|_Z is nondegenerate.  This is easy: given v \in T_pZ, the Kaehler condition gives \omega(v, Jv) > 0.

The monotone-implies-Fano direction is less elementary.  It relies on the Nakai-Moishezon-Kleiman criterion (see Lazarsfeld’s first positivity book):

Theorem.  Let L be a line bundle on a projective scheme X.  Then L is ample if and only if \int_V c_1(L)^{\text{dim}(V)} > 0 for every positive-dimensional irreducible subvariety V \subset X.

So, let (X, \omega, J) be a projective monotone Kaehler manifold.  To show that (X, J) is Fano it is enough to show that for every positive-dimensional complex submanifold Y^l, \int_Y c_1( K_X^{-1} ) > 0.  But c_1(K_X^{-1}) = c_1(TX) = \lambda[\omega_X], so this integral is equal to \lambda^l \int_Y \omega_X^l, 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 (M, \omega, J) is symplectic Fano if \langle c_1(TM), A\rangle > 0 for every nonzero A \in H_2(M; \mathbb{Z}) that can be represented by a J-curve.  Clearly monotone implies symplectic Fano, but the converse is false: e.g. there are 2-dimensional complex tori \mathbb{C}^2/\Lambda that have no holomorphic curves (hence are symplectic Fano) but are not projective (hence are not Fano) and do not admit an i-compatible monotone symplectic form (thanks John!).



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