exponential sum over variety I am wondering where to find a good reference for bounds of the type 
$$\sum_{x\in V(\mathbb{F}_p)} \chi(g(x))\psi(f(x))$$
where $V$ is a variety, $\chi$ is a multiplicative character over $\mathbb{F}_p$ and $\psi$ is an additive character over $\mathbb{F}_p$. 
What conditions for V, $f$, $g$ gives what kind of bounds? 
When $\chi$ or $\psi$ is trivial, what are known? Similarly, for nontrivial characters, what are known? 
 A: This is a very general sort of problem and you will get very different answers depending on exactly which $V, f,g$ you need. 
The least savings you could ask for is $\sqrt{p}$ savings over the trivial bound, i.e $O(p^{\dim V- 1/2})$. This essentially always applies, and can be proved without reference to etale cohomology theory, by reducing to the one-dimensonal case - see the paper of Weil.
The most savings you could hope for in any nontrivial case is square-root cancellation, i.e. $O(p^{ \dim V/2})$. This is expected to happen for most $V, f,g$, but nothing approaching a classification of cases where it fails is known. Instead we know a number of different results giving specific conditions, usually involving some kind of nonsingularity at infinity, that imply square-root cancellation. 
Some of the most general work in this area is due to Katz. The $\chi$ trivial case is studied here. The case $V = \mathbb A^n$ is studied here. I think you can find more references by following the links in those. If your case is similar to these but is not exactly covered, then perhaps the methods can be modified, and you can always use slicing to reduce to a lower-dimensional case at the cost of a somewhat weaker bound.
Often in practice you end up with a sum of this type where the polynomials do not satisfy any kind of nonsingularity condition but instead have other kinds of structure (e.g. multiplicative or additive symmetries). Then one could try to bring to bear any of the many tools of $\ell$-adic sheaf and etale cohomology theory to the problem. This is hard to summarize as it can get very messy.
