On higher dimensional extremal varieties of general type

TL;DR

This paper studies higher-dimensional analogues of Horikawa surfaces, called Horikawa varieties, focusing on their geometric, topological, and deformation properties, especially in extremal cases where the canonical volume equals twice the difference of the geometric genus and dimension.

Contribution

It introduces and analyzes Horikawa varieties, establishing new bounds, structure theorems, and properties related to their pluriregularity, fundamental groups, and deformations, extending classical surface results to higher dimensions.

Findings

Bounded geometric genus for singular image Horikawa varieties.

Horikawa varieties are simply connected despite singularities.

Optimal projective normality results for pluricanonical systems.

Abstract

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with inthe works of Fujita and Kobayashi. We carry out further studies of Horikawa varieties, proving new results on various geometric and topological issues concerning them. In particular, we prove that the geometric genus of those Horikawa varieties whose image under the canonical map is singular is bounded. We give an analogous result for polarized hyperelliptic subcanonical varieties, in particular, for polarized Calabi-Yau and Fano varieties. The pleasing numerology that emerges puts Horikawa's result on surfaces in a broader perspective. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected, even if $Y$ is singular. We also prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus $2$.