I've been hesitating about whether or not to answer this question. Not because it's uninteresting, but because I think it's too interesting. I'm glad you didn't ask what the influence of Frobenius is on characteristic $0$ geometry, because it's enormous and would be too much to get into. Since some of these have been discussed in other people's answers and elsewhere, I'll just make a list. These include: Mori's "bend and break", the theory of weights, the decomposition theorem for perverse sheaves, Frobenius splitting, Deligne-Illusie splitting, tight closure...

I'd rather get off the beaten track a bit, and discuss some answers which are more obscure, but hopefully interesting. Actually, these are closer to answers to the question you did ask.

1. Some special classes of complex varieties (e.g. toric varieties, modular curves, CM elliptic curves) carry endomorphisms or self correspondences which are in fact lifts of the Frobenius. For $\mathbb{P}^n$, can take $[x_0,\ldots x_n]\mapsto [x_0^p,\ldots x_n^p]$. Unfortunately, this usually doesn't happen. (I'd be happy with more examples, if people know of any.)

2. On K-theory the Adams operations $\psi^p:K^0(X)\to K^0(X)$ behave like the action of Frobenius. Defining these is a long story, suffice to say that for a line bundle $\psi^p(L) = L^p$, exactly like the Frobenius like endomorphism.

3. This is a bit of a crazy thing to do, but you can take the ultraproduct of the mod $p$ reductions of your complex variety and you will get a nonnoetherian (!) scheme dominating $X$ with a Frobenius.

As I said in my comment above, there are many answers. The best one really depends on what you want to do.

I've been hesitating about whether or not to answer this question. Not because it's uninteresting, but because I think it's too interesting. I'm glad you didn't ask what the influence of Frobenius is on characteristic $0$ geometry, because it's enormous and would be too much to get into. Since some of these interactions have been discussed in other people's answers and elsewhere, I'll just make a list. These include: Mori's "bend and break", the theory of weights, the decomposition theorem for perverse sheaves, Frobenius splitting, Deligne-Illusie splitting, tight closure...

I'd rather get off the beaten track a bit, and discuss some answers which are more obscure, but hopefully interesting. Actually, these are closer to answers to the question you did ask.

1. Some special classes of complex varieties (e.g. toric varieties, modular curves, CM elliptic curves) carry endomorphisms or self correspondences which are in fact lifts of the Frobenius. For $\mathbb{P}^n$, one can take $[x_0,\ldots x_n]\mapsto [x_0^p,\ldots x_n^p]$. Unfortunately, such lifts usually don't exist. (I'd be happy with more examples, if people know of any.)

2. On K-theory the Adams operations $\psi^p:K^0(X)\to K^0(X)$ behave like the action of Frobenius. Defining these is a long story, suffice to say that for a line bundle $\psi^p(L) = L^p$, exactly like the Frobenius.

3. This is a bit of a crazy thing to do, but you can take the ultraproduct of the mod $p$ reductions of your complex variety and you will get a nonnoetherian (!) scheme dominating $X$ with a Frobenius like endomorphism.

As I said in my comment above, there are many answers. The best one really depends on what you want to do.