  We study the global behavior of solutions to the nonlinear generalized
Hartree equation, where the nonlinearity is of the non-local type and is
expressed as a convolution, $$ i u_t + \Delta u + (|x|^{-(N-\gamma)} \ast
|u|^p)|u|^{p-2}u=0, \quad x \in \mathbb{R}^N, t\in \mathbb{R}. $$ Our main goal
is to understand behavior of $H^1$ (finite energy) solutions of this equation
in various settings. In this work we make an initial attempt towards this goal.
We first investigate the $H^1$ local wellposedness and small data theory. We
then, in the intercritical regime ($0<s<1$), classify the behavior of $H^1$
solutions under the mass-energy assumption $\mathcal{ME}[u_0]<1$, identifying
the sharp threshold for global versus finite time solutions via the sharp
constant of the corresponding convolution type Gagliardo-Nirenberg
interpolation inequality (note that the uniqueness of a ground state is not
known in the general case). In particular, depending on the size of the initial
mass and gradient, solutions will either exist for all time and scatter in
$H^1$, or blow up in finite time or diverge along an infinity time sequence. To
either obtain $H^1$ scattering or divergence to infinity, in this paper we
employ the well-known concentration compactness and rigidity method of
Kenig-Merle [36] with the novelty of studying the nonlocal nonlinear potential
given via convolution with negative powers of $|x|$ and different, including
fractional, powers of nonlinearities.
