Vanishing theorems on (l|k)-strong Kaehler manifolds with torsion

We derive sufficient conditions for the vanishing of plurigenera, $p_m(J), m>0$, on compact (l|k)-strong, $\omega^l\wedge \partial\bar\partial \omega^k=0$, Kaehler manifolds with torsion. In particular, we show that the plurigenera of compact (l|k)-strong manifolds, k<n-1, for which the holonomy of the unique Hermitian connection with skew-symmetric torsion is contained in SU(n) vanish. As a consequence all generalized k-Gauduchon manifolds with holonomy of the Hermitian connection with skew-symmetric torsion contained in SU(n) do not admit holomorphic (n,0) forms. Furthermore we show that all conformally balanced, (l|k)-strong Kaehler manifolds with torsion, k<n-1, are K\"ahler. We also give several compact examples of (l|k)-strong Kaehler and Calabi-Yau manifolds with torsion.