Given a positive integer
n
n
and a compact connected Riemann surface
X
X
, we prove that the symmetric product
S
n
(
X
)
S^n(X)
admits a Kähler form of nonnegative holomorphic bisectional curvature if and only if
genus
(
X
)
≤
1
\text {genus}(X)\, \leq \, 1
. If
n
n
is greater than or equal to the gonality of
X
X
, we prove that
S
n
(
X
)
S^n(X)
does not admit any Kähler form of nonpositive holomorphic sectional curvature. In particular, if
X
X
is hyperelliptic, then
S
n
(
X
)
S^n(X)
admits a Kähler form of nonpositive holomorphic sectional curvature if and only if
n
=
1
≤
genus
(
X
)
n\,=\,1\, \leq \, \text {genus}(X)
.