  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 $2$.
