ag.algebraic-geometry
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?)
I don't think the answer to the first question is known.
Will has already pointed out the trivial answer to the second question. However this is not the right question. I mean this is kind of trivial. The interesting question is if you fix the genus and require that the curve over $K$ has good reduction everywhere (outside a fixed set of primes). If you ask it in that way, then the answer for curves is negative (by Faltings) and so the easy fix to do it in higher dimensions does not work.
Here are some comments and references to Theorems 1,2,3:
Theorem 1 is known in more general context.
It does not need "strong", non-isotrivial is enough.
Relevant references are:
Kovács, Sándor J.(1-UT) Smooth families over rational and elliptic curves. J. Algebraic Geom. 5 (1996), no. 2, 369–385.
Kovács, Sándor J.(1-MIT) On the minimal number of singular fibres in a family of surfaces of general type. J. Reine Angew. Math. 487 (1997), 171–177.
Kovács, Sándor J.(1-CHI) Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom. 9 (2000), no. 1, 165–174.
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10 (2001), no. 4, 781–799.
Kovács, Sándor J.(1-WA) Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math. 131 (2002), no. 3, 291–317.
There are also generalizations for families over higher dimensional bases. See for instance:
Viehweg, Eckart(D-ESSN); Zuo, Kang(PRC-CHHK) Base spaces of non-isotrivial families of smooth minimal models. Complex geometry (Göttingen, 2000), 279–328, Springer, Berlin, 2002.
Kebekus, Stefan(D-KOLN); Kovács, Sándor J.(1-WA) Families of canonically polarized varieties over surfaces. (English summary) Invent. Math. 172 (2008), no. 3, 657–682.
Kebekus, Stefan(D-FRBG); Kovács, Sándor J.(1-WA) The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1–33.
Patakfalvi, Zsolt(1-PRIN) Viehweg's hyperbolicity conjecture is true over compact bases. (English summary) Adv. Math. 229 (2012), no. 3, 1640–1642.
Theorem 2:
This is a triviality unless you fix some invariants. On the other hand for relative dimension $1$ and fixed genus this is not true. This is the geometric version of Shavarevich's conjecture and was first proved by Parshin:
Paršin, A. N. Algebraic curves over function fields. I. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 1968 1191–1219,
and then in a more general case by Arakelov:
Arakelov, S. Ju. Families of algebraic curves with fixed degeneracies. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293.
In higher dimensions, the statement is true indeed by taking the product of an arbitrary family of curves and an arbitrary curve (each of the appropriate genus). The second curve can be moved in moduli which gives even a continuous family of families.
In fact, this was what led to the notion of strong isotriviality.
Some relevant references are:
Kovács, Sándor J.(1-WA) Strong non-isotriviality and rigidity. Recent progress in arithmetic and algebraic geometry, 47–55, Contemp. Math., 386, Amer. Math. Soc., Providence, RI, 2005.
Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
Zsolt Patakfalvi Arakelov-Parshin rigidity of towers of curve fibrations, connections to the infinitesimal Torelli problem http://arxiv.org/abs/1010.3069
Theorem 3:
as I explained above, even this is not true without the "strong" assumption.
For strongly non-isomorphic families it is proven in
Kovács, Sándor J.(1-WA); Lieblich, Max(1-WA) Boundedness of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture. (English summary) Ann. of Math. (2) 172 (2010), no. 3, 1719–1748.
I would expect it to be true for a somewhat larger class of families, but the actual class still needs to be defined. The key modulo this paper is rigidity.
For more details see
Kovács, Sándor J.(1-WA) Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture. Algebraic geometry—Seattle 2005. Part 2, 685–709, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009.
or
Chapter III of Hacon, Christopher D.(1-UT); Kovács, Sándor J.(1-WA) Classification of higher dimensional algebraic varieties. Oberwolfach Seminars, 41. Birkhäuser Verlag, Basel, 2010. x+208 pp. ISBN: 978-3-0346-0289-1
This is not a complete answer, but it's way too long for a comment.
You can define ampleness for arbitrary coherent sheaves. This is done for example in V. Ancona, "Faisceaux amples sur les espaces analytiques". The definition is the same as in the locally free case: A coherent sheaf $\mathscr E$ on a compact complex space $X$ (say, nonzero and torsion-free) is said to be ample if for any coherent sheaf $\mathscr F$ on $X$ there exists $n_0$ such that for $n \geq n_0$ the sheaf $\mathscr F \otimes S^n \mathscr E$ is globally generated.
This is equivalent to $\mathscr F \otimes S^n \mathscr E$ having vanishing higher cohomology groups for $n \gg 0$, and to the line bundle $\mathscr O_{\mathbb P(\mathscr E)}(1)$ on $\mathbb P_X(\mathscr E)$ being ample (Proposition 2.5 in Ancona's paper).
Now a nonzero quotient of an ample vector bundle will be ample in this general sense. Hence your question becomes: if $T_X$ contains an ample subsheaf $\mathscr F$, then is $X$ isomorphic to a projective space? In case the Picard number $\rho(X) = 1$, this is Corollary 4.3 in "Galois coverings and endomorphisms of projective varieties" by Aprodu, Kebekus, and Peternell. In fact they show that in this case, $\mathscr F$ is already locally free and then they apply the result of Andreatta and Wiśniewski.
Best Answer
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.