Rational curves on varieties of general type

Question

Let $S$ be a complex surface of general type. Are there infinitely many smooth rational curves on $S$? And more general, what if $V$ is a variety of general type?

Answer 1

I think that the best result in this direction is the following theorem due to Bogomolov:

Theorem. Let $S$ be a surface of general type with $c_1^2(S) > c_2(S)$. Then for any $g$ the curves of geometric genus $g$ on $S$ form a bounded family.

In particular, since a surface of general type cannot be covered by rational or elliptic curves, these curves cannot deform. So Bogomolov's result implies that if $c_1^2(S) > c_2(S)$ then $S$ contains only finitely many rational or elliptic curves.

In general, it is conjectured than rational curves are never Zariski dense on a variety $V$ of general type, and more precisely it is expected that they are contained in a proper subvariety (hyperbolicity conjecture).

If $\dim V \geq 3$, you can obviously have infinitely many of them: for instance, take $V= S\times C$, where $S$ is a surface of general type containing a smooth rational curve and $C$ is a curve of genus at least $2$.

Answer 2

I am surprised nobody mentioned the result of Lu and Miyaoka (Math. Res. Letter 2, 663-676 (1995)) which implies indeed that there are only finitely many smooth rational curves on a surface of general type, thus answering the question of the OP.

Answer 3

The conjecture Francesco is referring to as the "hyperbolicity conjecture" is actually the Green-Griffiths-Lang conjecture. It states that on any given smooth projective manifold of general type $X$ there should exist a proper subvariety $Y\subsetneq X$ such that for all non-constant holomorphic map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$.

This would imply in particular that the same proper subvariety should contain every rational or elliptic curve (or, more generally, every image of a complex torus).

Beside the result of Bogomolov (which treats "only" the algebraic part of this conjecture) one should cite also Mc Quillan's theorem, which prove this conjecture for surfaces under the same assumption on the second Segre number $c_1^2-c_2>0$.

This condition is a technical hypothesis which guarantees the existence of an algebraic (multi)foliation on the surface. The core of Mc Quillan's proof is then in showing that an algebraic (multi)foliation on a surface of general type does not admit any dense parabolic leaf.

In higher dimensions, very little is known even for the algebraic part of the conjecture, apart from the case of generic complete intersections of high (multi)degree (Clemens, Ein, Voisin, Pacienza...).

To finish with, thanks to very recent results of Cano about resolution of singularities of holomorphic foliations by curves on threefolds, probably Mc Quillan will be soon able to improve his previous result with the weaker assumption $13c_1^2-9c_2>0$.

Answer 4

Some related additional comments:

Siu-Yau showed that when the bisectional curvature of a compact K\"ahler variety is positive then there exists a rational curve.

Birkar, et al, showed that when $X$ is a Moishezon manifold which is not projective. Then $X$ contains a rational curve. Moreover we have the same result when $X$ is a normal Moishezon variety with analytic $\mathbb Q$-factorial singularities.

Existence of rational curve is related to Gromov-Witten Invariants de to Ruan-Tian.

Let $X$ be a smooth projective variety over $\mathbb C$. Fix a point $x\in X$, a homology class $A\in H_2(X(\mathbb C),\mathbb Z)$ and very ample divisors in general position $H_i\subset X$, $i=1,...,k$.

Let $y_0,...,y_k\in \mathbb CP^1$ be general points. Take maps $f:\mathbb CP^1\to X$ such that $f_*[\mathbb CP^1]=A$, $f(y_0)=x$ and $f(y_i)\in H_i$

Ruan-Tian, introduced an invariant $$\tilde F_{A,X}(x,H_1,...,H_k;y_0,...,y_k):={\text{number of such maps}}$$ Gromov-Witten theory of pseudo-holomorphic curves shows that one can make a similar definition where $X$ is replaced by a symplectic manifold $(M,\omega)$. Then the corresponding invariant is denoted by $$\tilde \Phi_{A,X}(x,H_1,...,H_k;y_0,...,y_k):={\text{number of such maps}}$$

Ruan-Tian, showed that if $\tilde \Phi_{A,X}(x,H_1,...,H_k;y_0,...,y_k):={\text{number of such maps}}$ then there is a rational map $f:\mathbb CP^1\to X$ such that $f_*[\mathbb CP^1]=A$, $f(y_0)=x$ and $f(y_i)\in H_i$

Moreover, $\tilde \Phi_{A,X}(x,H_1,...,H_k;y_0,...,y_k)=\tilde F_{A,X}(x,H_1,...,H_k;y_0,...,y_k)$ if the following conditions are satisfied.

• If $C_1,...,C_m\subset X$ are rational curves such that $\sum [C_i]=A$ and $x\in C_1$, then $m=1$

• If $g:\mathbb CP^1\to X$ is any map such that $g_*[\mathbb CP^1]=A$, $g(y_0)=x$ then $H^1(\mathbb CP^1,g^*TX)=0$

Existence of rational curves are very important in the classification of the solution of K\"ahler Ricci flows .

i.e an effective way to consider Green-Griffiths-Lang conjecture is using classification of the solutions of relative Kahler Ricci flow and study of existence of rational curve appears,