  In this paper, we consider the global existence and blowup phenomena of the
following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=\Delta
u-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0,
  &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*}
where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is
even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$
is a complex-valued function of $x$. We obtain some sufficient conditions and
establish two sharp thresholds for the blowup and global existence of the
solution to the problem. These results can be looked as the supplement to
Chapter 6 of \cite{Cazenave2}. In addition, our results extend those of
\cite{Zhang} and improve some of \cite{Tao2}.
