I will essentially say the same thing as in John's answer, but in a different way, and maybe this will be easier to understand (judge for yourself).
"Global," here, does not mean global variables, it means that you need to know more than just the geometry local to the event horizon to know that you are at an event horizon. Specifically, you need to know that there is a singularity somewhere else (e.g., at $r=0$ in a Schwarzschild spacetime).
If you are at some point $x^\mu$ in spacetime, and you need to know properties of the spacetime non-local to $x^\mu$ in order to infer property $X$, then $X$ is a global feature of the spacetime. Thus, event horizons are a global feature of your geometry.
Another way to understand this is to recognize that the event horizon is not a geometrically significant point in your spacetime. It makes no sense to talk about the event horizon unless you add the concept of causality to your physics. Contrast this to the singularity, which by Einstein's theory is explicitly intrinsically geometric in nature. Thus, if you have a metric and change your system of coordinates, then no change will disguise the breakdown of your metric at the singularity, since it is a geometric feature, but the event horizon has no such privilege: spacetime is smooth at the event horizon, and will look like any other patch of spacetime (assuming you can see neither the effects of the black hole singularity "nearby" nor the AMPS firewall, which may or may not exist...). Hence, given a black hole spacetime in some arbitrary system of coordinates, it may not be easy to spot that some particular point $r_X$ defined in the metric somewhere is an event horizon, not at first glance anyway.
Finally, to understand how my answer is, really, identical to John's answer it simply needs to be made manifest that time, obviously, is a spacetime coordinate. Thus talking of "globally geometric properties of your spacetime" is really just saying "future histories" in a different way.