I am reading the book "Calabi-Yau Manifolds: A Bestiary for Physicists" by Tristan Hübsch, and on pages 30-31 (for an excerpt, please see https://books.google.com/books?id=waPsCgAAQBAJ&lpg=PP1&pg=PA30#v=onepage&q&f=false; unfortunately page 31 isn't part of this excerpt) the author first explains the isomorphisms
$$\cdot\Omega: T_M \xrightarrow{\sim} \Lambda^2 T_M^* \qquad (1)$$ $$\cdot\Omega: H^{q}(M, T_M) \xrightarrow{\sim} H^{2,q}(M)\qquad (2)$$
where $\Omega$ is the holomorphic $(3,0)$ form for the Calabi-Yau 3-fold.
He then says that since the canonical bundle of the CY 3-fold is trivial ($\mathcal{K}_M = \mathbb{C}_M$), one can drop it from the Serre Duality formula:
In view of this, the usual Serre duality formula
$$H^{q}(M, \mathcal{V})^\star \approx H^{3-q}(M, \mathcal{V}^* \otimes \mathcal{K}_{M})\qquad (3)$$
simplifies in that ``$\otimes \mathcal{K}_M$'' may be omitted. For example, $$H^{q}(M, T_M)^\star \approx H^{3-q}(M, T_M^*)\qquad (4)$$ which together with (2) implies $$H^{2,q}(M)^\star \approx H^{3-q}(M, T_M^*) \equiv H^{1,3-q}(M) \quad (5)$$
I am happy so far. But the next statement confuses me:
Furthermore, since $H^{3,0}(M) \approx \mathbb{C}$, there is a new duality, implying $$H^{p,0}(\mathcal{M})^\star \approx H^{3-p,0}(M) \qquad (6)$$
How does equation (6) follow from equations (1)-(5)? I am sure this is something very elementary which I am missing :-(
My question assumes that (6) is inferrable using (1)-(5). Is there additional information needed?