Remarks on polynomial count varieties

TL;DR

This paper investigates polynomial count varieties, showing that smooth polynomial count varieties with specific point counts are not necessarily affine spaces, and their Hodge numbers do not always satisfy certain symmetry conditions.

Contribution

It provides counterexamples to common assumptions about the structure and Hodge numbers of polynomial count varieties.

Findings

Smooth polynomial count varieties with X(q)=q^n are not necessarily affine spaces.

Polynomial count varieties do not always have Hodge numbers satisfying p=q in fixed weight.

Answers to natural questions about the structure of polynomial count varieties are negative.

Abstract

In this short note we prove a couple of facts about polynomial count varieties, answering natural questions that they raise. A polynomial count $X$ variety is essentially one for which its number of points over finite fields is given by a polynomial in the field size. Well-known examples include affine or projective space (or more generally the Grassmanian) and other standard varieties. The two questions we address are the following. 1) If $X$ is smooth, polynomial count with $\#X(q)=q^n$ for some $n$, is $X$ isomorphic to $n$-dimensional affine space? 2) If $X$ is a polynomial count, is it true that its Hodge numbers in a given graded piece of fixed weight satisfy~$h^{p,q}=0$ unless $p=q$? We show that in both cases the answer is no.