A very long-standing problem in Algebraic Geometry is to determine
the stability of exceptional bundles on smooth projective
varieties.

We first show that if $X\subset\Pn$, $n\geq 3$, is a smooth and irreducible hypersurface of degree
$1\leq d\leq n-1$ and $E$ is an exceptional bundle on $X$ given by
\[ 0 \rightarrow \mathcal{O}_{X}(-1)^{r} \rightarrow
\mathcal{O}_{X}^{s} \rightarrow E \rightarrow 0, \]
for some $s,t\geq 1$, then $E$ is stable.

We then prove that
any exceptional bundle on a smooth complete intersection
$3$-fold $Y\subset\mathbb{P}^n$ of type $(d_1,\ldots,d_{n-3})$ with
$d_1+\cdots+ d_{n-3}\leq n$ and $n\geq 4$, is stable.

This is joint work with Rosa Maria Mir\'{o}-Roig.