Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type whose image of its canonical map is of maximal dimension, satisfies $\textrm{Vol} \geq 2 (p_{g} - n)$. In this article, we investigate the very interesting extremal situation of varieties with $\textrm{Vol} = 2(p_{g} - n)$, which we call Horikawa varieties for they are natural higher dimensional analogues of Horikawa surfaces.

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. We prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2.

Even though there are infinitely many families of Horikawa varieties in any given dimension $n$, we show that when the image of the canonical map is singular, the geometric genus of the Horikawa varieties is bounded by $n + 4$.