Fixed Point Property in Algebraic Geometry

I am wondering about the following problem: for which (say smooth, complex, connected) algebraic varieties $X$ does the statement any regular map $Xto X$ 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?

• Next How do you find the potential function V of the gradient system?
• Previous Suggest effective heuristic (not precise) graph colouring algorithm