There's also Fujita's conjecture.
Conjecture: Suppose $X$ is a smooth projective dimensional complex algebraic variety with ample divisor $A$. Then
- $H^0(X, \mathcal{O}_X(K_X + mA))$ is generated by global section when $m > \dim X$.
- $K_X + mA$ is very ample for $m > \dim X + 1$
It's also often stated in the complex analytic world.
Also there are many refinements (and generalizations) of this conjecture. For example, the assumption that $X$ is smooth is probably more than you need (something close to rational singularities should be ok). It also might even be true in characteristic $p > 0$.
It's known in relatively low dimensions (up to 5 in case 1. I think?)