Are there nontrivial continuous maps between complex projective spaces? Are there maps $f: \Bbb{CP}^n \rightarrow \Bbb{CP}^m$, with $n>m$, that are not null-homotopic? In particular, is there some non-null-homotopic map $\Bbb{CP}^n \rightarrow S^2$ for $n>1$? Can we say anything more generally about the pairs $(n,m)$ that admit such maps?
Note that, in the real case, the answer to the first quesiton is yes and the second no: the Hopf map descends to a non-null-homotopic map $\Bbb{RP}^3 \rightarrow \Bbb{RP}^2$, and there's a natural bijection between $[\Bbb{RP}^n, S^1]$ and $H^1(\Bbb{RP}^n; \Bbb Z) = 0$.
This question asks about the $n \leq m$ situation.
Edit: Jason DeVito's post below answers both questions in the affirmative; there remains, though, the following interesting case:
Is there a map $f: \Bbb{CP}^2 \rightarrow \Bbb{CP}^1$ that is not null-homotopic?

I seem to be able to argue via obstruction theory that there is no nontrivial map $\mathbb{C} P^2\to S^2$. That is, taking any map $f$ on the 2-skeleton $\mathbb{C}P^1\subset \mathbb{C}P^2$, $f$ extends to the 4-skeleton iff the cellular cochain that takes the unique 4-cell $e_4$ to the homotopy class of $f|_{\partial e_4}$ is trivial in $C^4(\mathbb{C} P^2;\pi_3(S^2))$. If $f$ is of degree $1$, then $f(e_4)=1\in\pi_3(S^2)$, where $1$ is the Hopf map, since the 4-cell of $\mathbb{C} P^2$ is attached via the Hopf map. Then if $f$ is of degree $n$, it maps the 4-cell to $n$. Thus the obstruction is insurmountable except in case $n=0$. 
The problem with this argument is that it will show that there are no maps from $\mathbb{C}P^n$ to $\mathbb{C}P^1$ for any $n>1$, and Jason deVito's comment to his first answer indicates there is one for $n=5$.
