In the first part of this work (sects. 1-3) we consider an irreducible normal variety $V_{3}$ of dimension 3 in a complex projective space. Let $p_{a}(V3)$ and $P_{a}(V3)$ be the virtual arithmetic genus and the second arithmetic genus of $V_3$ respectively. We prove that the equality $p_{a}(V3) = V_{a}(V3)$ holds if and only if $V_{3}$ is Cohen-Macaulay. As previously remarked in [11], we obtain the relation $P_{a}(V3) \ge p_{a}(V3)$ for any normal $V_{3}$. We also give an example of $V_{3}$’s on which the inequality $P_{a}(V_{3}) > p_{a}(V_{3})$ holds. The problems we treat here are strictly close to some arguments geometrically developed by Marchionna in [11]. In the second part (sec. 4) we consider a normal algebraic variety $V_{d}$ of dimension $d \ge 2$, in a complex projective space. Suppose $V_{d}$ has multiple subvarieties of dimension at most $k(k \le d — 2)$. By employing a theorem due to Grauert-Riemenschneider, we prove that $H^i(V_{d},\omega_{V_{d}} \bigotimes \mathcal{L}) = o, (i > k)$, where $\omega_{V_{d}}$ denotes the dualizing sheaf, and $\mathcal{L}$ is an ample (invertible) sheaf on $V_{d}$. This fact implies the strong theorem on the regularity of the adjoint on a normal variety $V_{d}$ with isolated singularities.