I am wondering about the following problem: for which (say smooth, complex, connected) algebraic varieties $X$ does the statement *any regular map $X\to X$ has a fixed point* hold?
MathSciNet search does not reveal anything in this topic.

This is true for $\mathbb{P}^n$ (*because its cohomology is $\mathbb{Z}$ in even dimensions
and $0$ otherwise, and the pullback of an effective cycle is effective, so all summands
in the Lefschetz fixed point formula are nonnegative, and the 0-th is positive* -- is this a correct argument?). Is it true for varieties with cohomology generated by algebraic cycles (i.e. $h^{p,q}(X)=0$ unless $p=q$ and satisfying Hodge conjecture), for example for **Grassmannians**, **toric varieties**, etc.? This is not at all clear that the traces of $f$ on cohomology will be nonnegative.

Probably you have lots of counterexamples. What about positive results?