Varieties isogenous to a higher product with prescribed numerical invariants

TL;DR

This paper proves a nonexistence result for varieties isogenous to a higher product of dimension $n\ge 4$ with fixed topological Euler number $e(X)=(-2)^n$ and vanishing first Betti number. By combining the Euler relation $e(X)=\frac{(-2)^n}{|G|}\prod (g(C_i)-1)$ with $b_1(X)=0$ forces $|G|=\prod (g(C_i)-1)$ and $C_i/G\cong \mathbb{P}^1$; it then analyzes possible ramification structures via spherical systems of generators, bounding $r(i)$ and the associated invariants $\Theta(A_i)$ and $\alpha(A_i)$. Through a sequence of dimension bounds and a exhaustive group-theoretic elimination (leveraging classifications of automorphism groups of low-genus surfaces), it is shown that no finite group $G$ can satisfy all necessary conditions when $n\ge 4$, although examples exist for $n=2,3$. Consequently, there are no fake $\mathbb{P}^1(\mathbb{C})^n$ in dimension $n\ge 4$ arising from varieties isogenous to a higher product with the prescribed invariants. The results illuminate the interplay between group actions on products of curves and topological invariants of quotients, constraining the landscape of possible higher-product varieties of general type.

Abstract

Using structural properties of groups of small order, we establish the non-existence of varieties isogenous to a higher product of dimension $n$ greater than 3 with fixed topological Euler number $(-2)^n$ and trivial first Betti number.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof