I am studying the String Theory book by Becker, Becker, and Schwarz, and I decided to verify the Hodge diamonds for a CY3 and a CY4. These can be found on page 365 and they are eq.(9.14) and (9.16). It was very easy for me to derive the precise form of the Hodge diamond following what they mention in the book. A problem arose when I tried to repeat the computation for the case of a CY4.
Let me first state the problem/question. Why for a CY4 the Hodge number $h^{2,0}$ is equal to zero? This is not obvious to me from the relations they give in the book.
Allow me to show you how I have worked out the rest of the elements of the Hodge diamond.
Let me write here the properties the book gives for the Hodge numbers. For a Calabi-Yau n-fold we have that -these are eq.(9.10)-(9.12) in the book
$\begin{equation}
\begin{split}
h^{p,0} &= h^{n-p,0} \\
h^{p,q} &= h^{q,p} \\
h^{p,q} &= h^{n-q,n-p}
\end{split}
\end{equation}$
and we know that for a simply connected manifold $h^{1,0}=h^{0,1}=0$ and that a compact connected Kahler manifold has $h^{0,0}=1$.
From the first of the above properties, we have the following relations
$\begin{equation}
\begin{split}
&h^{4,3} = h^{3,4}, \qquad h^{4,2} = h^{2,4}, \qquad h^{4,1} = h^{1,4}, \qquad h^{4,0} = h^{0,4}, \qquad \qquad h^{3,2} = h^{2,3}, \qquad h^{3,1} = h^{1,3}, \\
&h^{3,0} = h^{0,3}, \qquad h^{2,1} = h^{1,2}, \qquad h^{2,0} = h^{0,2}, \qquad h^{1,0} = h^{0,1}
\end{split}
\end{equation}$
From the second property we have
$\begin{equation}
h^{4,0} = h^{0,0} \qquad h^{3,0} = h^{1,0}
\end{equation}$
And finally, from the third one we obtain
$\begin{equation}
h^{4,4} = h^{0,0}, \qquad h^{4,3} = h^{1,0}, \qquad h^{4,2} = h^{2,0}, \qquad h^{4,1} = h^{1,0}, \qquad h^{3,3} = h^{1,1}, \qquad h^{3,2} = h^{2,1}.
\end{equation}$
The undetermined $h^{2,2}$ is given by -see eq.(9.17)
$\begin{equation}
h^{2,2} = 2 (22+2h^{1,1}+2h^{1,3}-h^{1,2})
\end{equation}$
If we impose all of the above to the Hodge diamond we get precisely what is shown in the book with the only difference that in the book they seem to have that $h^{2,0}=0$ which I cannot obtain.
Best Answer
The answer depends on the definition of a CY variety that you are using.
If the definition is just the triviality of the canonical class and simply-connectedness then it is just not true that $h^{2,0}$ is zero. In fact, any hyperkahler fourfold is a counterexample (for instance, the Hilbert square of a K3-surface).
A more restrictive definition includes the assumption $h^{2,0} = 0$ (in fact, this is motivated by the Bogomolov Decomposition Theorem).
So, I guess, in your case the condition $h^{2,0} = 0$ holds by definition.