\[lem:extremalray\]). Then a symplectic Graber-Harris-Starr type result [@SymRC] will prove Koll[á]{}r’s conjecture in this special case. However, due to the fact that we are unable to compute all the Gromov-Witten invariants associated to the extremal rays, we need, in addition, a topological argument to show the fibration structure is preserved under symplectic deformation.

**Acknowledgement:** I would like to thank Tom Graber, Yongbin Ruan, Jason Starr, Claire Voisin, and Aleksey Zinger for discussions about various part of the project, Zhiyuan Li and Letao Zhang for their encouragement, and Chenyang Xu for providing the key argument in Lemma \[lem:structure\] using “fake projective spaces".

rational curves of small $-K_X$ degree {#sec:2}
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The main theorem of this section is the following.

\[thm:lowdegree\] Let $X$ be a smooth projective Fano $4$-fold of pseudo-index at least $2$ and $f: C\cong {{\mathbb}{P}}^1 \to X$ be a very free curve such that $-f^*K_X \cdot C$ is at most $8$. Then there is a non-zero Gromov-Witten invariant with two point insertions that counts the number of irreducible very free curves satifying the incidence relations.

The strategy of the proof is the following. We want to show that any curve (in the chosen class) that can meet all the 