TITLE: Rational curves on varieties of general type
QUESTION [14 upvotes]: 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?

REPLY [2 votes]: 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,