Standard integral Kähler form on $\mathbf{CP}^1 \times \mathbf{CP}^2$

Question

$\newcommand{\Proj}{\mathbf{CP}}$What is the "standard integral Kähler form" on $\Proj^1 \times \Proj^2$? Does that mean Fubini-Study form on $\Proj^1$ and $\Proj^2$？

Answer 1

$\newcommand{\Proj}{\mathbf{CP}}$I've never heard the term "standard integral Kähler form" applied to anything but a projective space, but probably your guess is correct: The phrase refers to the Kähler form of the product metric with integral Fubini-Study forms on each factor.

In detail, if $p_{1}$ and $p_{2}$ are the projections to the respective factors, and if $g_{1}$ and $g_{2}$ are the respective Fubini-Study metrics on $\Proj^{1}$ and $\Proj^{2}$ whose Kähler forms generate the respective integer cohomologies, then the "standard integral Kähler form" on the product probably refers to $\omega$, the Kähler form of $g = p_{1}^{*}g_{1} + p_{2}^{*}g_{2}$.