PomeExample 21.12.12.CD7BA5BA48AF Let us move on with our favorite example example 21.12.4. We continue to use the same notations. In this case,
\[ G_0(m)= \begin{cases} m\log m+(1-m)\log (1-m),& \textup{if } m\in [0,1],\\ \infty ,& \textup{otherwise}. \end{cases} \]
The Legendre transform is given by
\[ F_0(n)=\log \left(1+\mathrm{e}^ n\right). \]
Composing with the tropicalization map, we find that
\[ \omega |_{\mathbb {C}^*}(z)=\mathrm{dd}^{\mathrm{c}}\log \left(1+|z|^2\right). \]
This is exactly the Fubini–Study metric as we have seen in example 5.20.6.
We can now explain one subtlety: in our expression (21.12.9), there is no factor $1/2$ before the sum; the absence of the factor $1/2$ matches the choice $\operatorname{Trop}(z)=\log |z|^2$ rather than $\log |z|$ in our tropicalization map (21.12.3).
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.