How "frequent" are smooth projective varieties $X$ with (anti-)ample canonical bundle $\omega_X = \bigwedge^d \Omega^1_{X/k}$?

E.g. for curves $C/k$, the canonical bundle is ample iff the genus $g(C) > 1$ and anti-ample iff $g(C) = 0$. What is the situation like in the higher dimensional case?

(This question is inspired from How "frequent" are smooth projective varieties with trivial canonical bundle?)

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    $\begingroup$ The Wiki article misses some facts. Work of Koll\'ar-Miyaoka-Mori, after earlier work by Nadel, proves there are only finitely many deformation families of $n$-dimensional Fano manifolds for each $n$. Also there is a bijection between toric Fano manifolds and integral rational polytopes which are "reflexive", so classification of these comes to classifying reflexive polytopes. Among smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, or complete intersections of hypersurfaces of degrees $d_1,\dots,d_r$, the Fano condition is $d\leq n$, resp. $d_1 + \dots + d_r \leq n$. $\endgroup$ – Jason Starr Nov 25 '11 at 19:17

Just as in the case of curves, there is in general a trichotomy of cases: Let $X$ be a smooth projective variety. Then $X$ is built from pieces $Y$ with

  • $\kappa(Y)<0$

  • $\kappa(Y)=0$

  • $\kappa(Y)=\dim Y$.

Here $\kappa$ denotes the Kodaira dimension. If the anti-canonical is ample, i.e., $X$ is a Fano, then it belongs to the first class, if the (anti-)canonical is trivial it belongs to the second and if the canonical is ample, then it belongs to the third. Otherwise one can perturb these cases birationally or by taking a finite quotient to get other examples in these classes.

"Built" means the following: Any such $X$ is birational to $\widetilde X$ that has an iterated fiber space structure with general fiber a Fano variety (for each fiber space) and the target a variety that has $\kappa\geq 0$. Then this target has an iterated fiber space structure with general fiber $\kappa=0$ (for each fiber space) and target that has $\kappa=\dim$ (i.e., it is a variety of general type). (The first series of fibrations come from the MMP and they are sometimes called Mori fiber spaces (each step individually) and the second is the Iitaka fibration).

As far as how "frequent" the cases are, it seems that the curve case kind of tells us the relative frequency of each compared to the others. Of course we have less explicit data in higher dimensions, but as Jason (and Balázs answering the linked question) mentioned there are results to suggest that we should expect a similar distribution.

In the curve case it is interesting that the same trichotomy appears from various points of view and in some of these we can put some quantitative measure on their relative frequency:

  • Topology: the fundamental groups in each case: trivial, abelian, non-abelian

  • Arithmetic: the group of rational points in each case (this has to be taken with a grain of salt, but it is instructive): non-finitely generated, finitely generated, finite.

  • Differential Geometry: curvature in each case: positive, flat, negative or if you like parabolic, elliptic, hyperbolic.

  • Algebraic Geometry: the dimension of the moduli space in each case: $0$, $1$, $3g-3$

Now one can go out and try to work out what happens to these classifications in any of these disciplines. As far as I know what (little) evidence we have suggests that the same kind of ratio occurs always: The $\kappa<0$ case is relatively small, the $\kappa=0$ is a little larger, but even combined they are nowhere near the "frequency" of the last case, which is accordingly named general type.

One interesting thing is that even though about 100% of varieties is of general type, most of those that we know explicitly are not. This is not a contradiction though. The fact that we can easily write down a description of a variety either by their equation or by a construction makes them special, so it is not surprising that these are not "general".

At the same time, by a different measure rational curves are quite ubiquitous. Abstractly there is only one smooth projective rational curve, but it appears everywhere. If you consider any birational morphism between smooth varieties, then the exceptional divisors will always be covered by rational curves. So one might argue that they are quite frequent.

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    $\begingroup$ Obviously this is highly subjective, but I disagree with S\'andor's comment that "even though about 100% of varieties is of general type, most of those that we know explicitly are not." Most of those that we know explicitly are of general type: if you write down a random homogeneous polynomial, almost certainly the zero set is of general type. What is remarkable is that, although most varieties are of general type, the ones which are not nonetheless play a major role in mathematics: homogeneous spaces, moduli of genus 0 curves, moduli of vector bundles with fixed determinant on a curve ... $\endgroup$ – Jason Starr Dec 5 '11 at 11:38
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    $\begingroup$ Jason, I guess you are correct in a strict sense, but I don't know if we can really say that we know explicitly the zero set of a random polynomial. This is a minor philosophical point and my main point was exactly what you're saying at the end. I tried to say that in a concise way. Thanks for making my point explicit (!). :) $\endgroup$ – Sándor Kovács Dec 5 '11 at 17:07

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