📝 SummarXiv

Abstract

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If Standard conjecture D holds on $X\times X$, then all polarised endomorphisms on $X$ are semisimple. Result 2: Being an endomorphism of $X$, we can compose $Fr_X$ a \textbf{positive integer} of times, for example $Fr_X^2=Fr_X\circ Fr_X$, $Fr_X^3=Fr_X\circ Fr_X\circ Fr_X$. On the cohomological level, we can define $(Fr_X^*)^s$ for all integers $s$. What if we can define $(Fr_X^*)^s$ for all \textbf{real numbers} $s$, in a good way (to be made precise later)? This short note presents an approach towards so-called Dynamical degree comparison conjecture and Norm comparison conjecture (allowing to bound the growth of the pullback of iterations of an endomorphism on cohomology groups in terms of that on algebraic cycles), for dominant rational maps and more generally dynamical correspondences, proposed previously by Fei Hu and the author, via such a possibility. The main upshot is a heuristic argument to show that the mentioned conjectures should follow from Standard conjecture D. All this discussion also holds if we replace $Fr_X$ by another polarised endomorphism on $X$.