On projective varieties of general type with many global k-forms

Abstract: We prove the following results: (1) for any nonsingular projective $3$-folds $X$ of general type with $\chi(\mathcal{O}<em>X)\neq 2,3$, the canonical volume $\text{Vol}(X)$ has the optimal lower bound $\frac{1}{420}$; (2) for nonsingular projective $3$-folds (resp. $4$-folds) $X$ of general type with $h^{2, 0}(X)\geq 108\cdot 42<sup>3+4$ (resp. with sufficiently large $h^{2, 0}(X)$), the $3$-canonical map (resp. $5$-canonical map) is stably birational; (3) for any nonsingular projective $n$-fold $X$ of general type with $q(X)&gt;n\geq 4$, the canonical stability index $r_s(X)$ is upper bounded by the $(n-1)$-th canonical stability index $r</em>{n-1}$.