Let
X
⊆
P
N
X\subseteq {\mathbb {P}^{N}}
be a smooth variety with normal bundle
N
X
\mathcal {N}_X
. In this note we prove that if
N
X
⊗
O
P
N
(
−
t
)
\mathcal {N}_X\otimes \mathcal {O}_{{\mathbb {P}^{N}}}(-t)
is an instanton with quantum number
k
k
, then
dim
(
X
)
=
N
−
2
\dim (X)=N-2
,
t
=
1
t=1
and, when
dim
(
X
)
≥
4
\dim (X) \ge 4
, also
⌊
dim
(
X
)
/
2
⌋
≤
deg
(
X
)
\lfloor \dim (X)/2\rfloor \le \deg (X)
. Moreover, we also discuss some methods for constructing smooth varieties such that
N
X
⊗
O
P
N
(
−
1
)
\mathcal {N}_X\otimes \mathcal {O}_{{\mathbb {P}^{N}}}(-1)
is an instanton with
k
≠
0
k\ne 0
, illustrating them with explicit examples and counterexamples.