I read several books and articles about Yau's proof of Calabi's conjecture. I do want to understand how and why such evidence actually works, but most articles are for specialists and give little or no details at several stages.

For example, in Moroianu's Conferences on Kähler Geometry or the Einstein Manifolds in Besse, one begins with a compact Kähler connected manifold. $ (X ^ n, omega) $, or $ omega in mathcal {A} ^ {1, 1} (X) $ is his Kähler form. Then if $ nu_g $ is the Riemannian measure associated with $ X $, we define $$ mathcal {K}: = left { varphi in mathcal C ^ infty (X, mathbb R): omega + dd ^ c varphi> 0, int_X varphi , mathrm {d} nu_g = 0 right }. $$

After some considerations, we also define the set

$$ mathcal K: = left {f in mathcal C ^ infty (X, mathbb R): int_X e ^ f cdot omega ^ {{n)} = int_X omega ^ {(n)} right }, $$

or $ omega ^ {(n)} $ is the $ n $corner product $ omega $.

It turns out that proving Calabi's conjecture amounts to proving that the map $ mathrm {Cal} colon mathcal {K} to mathcal {K} $ is a diffeomorphism, where $$ mathrm {Cal} ( varphi): = log left ( frac {( omega + dd ^ c varphi) ^ {(n)}} { omega ^ {(n)}} right ). $$

In fact, this is a fairly standard procedure that appears almost everywhere in this area.

Now, the log bugs me a lot, what does it mean by the quotient $ displaystyle frac {( omega + dd ^ c varphi) ^ {(n)}} { omega ^ {(n)}} $?, and how does it make sense to take the "logarithm" of such a thing?

We also know that $ omega ^ {(n)} = n! mathrm {vol} $, or $ mathrm {vol} $ is the volume form of $ X $

And please, do not be so harsh with me, I'm still an undergraduate and I'm actually learning all of these things on my own. I want an answer as clear and detailed as possible, no matter if it's a trivial thing after all. Thanks in advance!