  Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \frac {\partial^2}{\partial
z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it
Hessian Nilpotent}(HN) if its Hessian matrix $\Hes P(z)=(\frac {\partial^2
P}{\partial z_i\partial z_j})$ is nilpotent. In recent developments in [BE1],
[M] and [Z], the Jacobian conjecture has been reduced to the following
so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any
homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $\Delta^m
P^{m+1}(z)=0$ for any $m>>0$.} In this paper, we first show that, the VC holds
for any homogeneous HN polynomial $P(z)$ provided that the projective
subvarieties ${\mathcal Z}_P$ and ${\mathcal Z}_{\sigma_2}$ of $\mathbb C
P^{n-1}$ determined by the principal ideals generated by $P(z)$ and
$\sigma_2(z):=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular
points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the
symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero
fixed point $w\in \mathbb C^n$ with $\sum_{i=1}^n w_i^2=0$. Secondly, we show
that the VC holds for a HN formal power series $P(z)$ if and only if, for any
polynomial $f(z)$, $\Delta^m (f(z)P(z)^m)=0$ when $m>>0$.
