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?



Fixed Point Property in Algebraic Geometry
By demand I expand a little on my answer. The holomorphic Lefschetz fixed point formula (aka the Woods-Hole formula) considers an endomorphism $f\colon M \to M$ of a smooth and compact complex manifold $M$ (or proper
smooth algebraic variety) with only isolated fixed points which are also assumed to be non-degenerate (i.e., the
tangent map of $f$ at a fixed point does not have eigenvalue $1$. Then the alternating trace of the action of $f$ on $H^\*(M,\mathcal O\_M)$ is equal to a sum over the fixed points $p$ of $1/det(1-T\_p(f)$. In particular if there
are no fixed points the alternating trace is equal to $0$. However, in the case when also $H^i(M,\mathcal O\_M)=0$ for $i>0$ then the alternating trace is equal to $1$ so the assumption that there are no fixed points gives a contradiction. Note, that the dimension of $H^i(M,\mathcal O\_M)$ is just $h^{0,i}$ so that an assumption that the Hodge numbers vanish off the diagonal gives the required vanishing by a good margin. Furthermore, the vanishing of just $h^{0,i}$ for $i>0$ is much much weaker, it is for instance a birational condition whereas blowing up a smooth curve of genus $>0$ in a variety of dimension at least 3 always give
off diagonal Hodge numbers.


As for the question of whether a birational involution of $\mathbb C^n$ always has a fixed point this seems trickier. It is true that the involution can be made to act regularly on a smooth and proper model and hence by the above has a fixed point. It is not clear however that the fixed point will map to a point of $\mathbb C^n$ as $\mathbb C^n$ is not proper.


