Suppose I have a smooth complete surface $S$ embedded in $\mathbf{P}^n$. Is there a known or conjectured lower bound (preferably sharp) on the degree of $S$ in terms of its numerical invariants?
In the case of curves this is the classical Castelnuovo bound (<http://en.wikipedia.org/wiki/Castelnuovo_bound>).



Embedding algebraic surfaces in projective space
Perhaps this paper of Harris is what you are looking for: [A bound on the geometric genus of projective varieties](http://www.numdam.org/item?id=ASNSP_1981_4_8_1_35_0). It generalizes Castelnuovo's bound to smooth projective varieties of arbitrary dimension. In page 44, we find the following statement: 



> 
> **Theorem.** Let $V$ be a non-degenerated smooth
>  and irreducible variety of dimension
>  $k$ and degree $d$ in $\mathbb P^n$.
>  If we set $M=\left[
> > \frac{d-1}{n-k}\right]$ then $$ 
> > h^0(V,\Omega^k\_V) \le \binom{M}{k+1} (n-k) + \binom{M}{k} (d-1-M(n-k)). $$
>  In particular, if $d \le k(n-k) + 1$ then $h^0(V, \Omega^k\_V)=0.$
> 
> 
> 


