Take the 2-minute tour ×  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?  share|improve
this question Footnote: "MathSciNet search does not reveal
anything in this topic" is not surprising, sin