= Solution
The restriction of $J$ to the $J$-invariant bundle $TY$ is an <almost complex manifold>[almost complex structure]. Its <Nijenhuis tensor> is the restriction of the ambient Nijenhuis tensor because vector fields tangent to an <embedded submanifold> have tangent Lie bracket. The ambient tensor vanishes since $X$ is a <complex manifold>, so the <Newlander-Nirenberg theorem> makes the induced structure on $Y$ integrable. The inclusion has complex-linear differential and is therefore holomorphic; hence $Y$ is a <complex submanifold>.
For a complex submanifold, the <holomorphic normal bundle> is
$$
N_{Y/X}=TX|_Y/TY.
$$
If $Y$ is a smooth hypersurface, taking top exterior powers in the <holomorphic conormal sequence>
$$
0\longrightarrow N_{Y/X}^*\longrightarrow\Omega_X^1|_Y\longrightarrow\Omega_Y^1\longrightarrow0
$$
gives
$$
K_X|_Y\cong N_{Y/X}^*\otimes K_Y.
$$
The normal bundle of a hypersurface is $N_{Y/X}\cong\mathcal O_X(Y)|_Y$, so the <Adjunction formula> is
$$
K_Y\cong(K_X\otimes\mathcal O_X(Y))|_Y.
$$
On $X=\mathbb{CP}^n\times\mathbb{CP}^m$, a bihomogeneous polynomial of bidegree $(d_1,d_2)$ is a section of the <holomorphic line bundle> $\mathcal O(d_1,d_2)$. Its zero locus is smooth precisely when the section is <transverse intersection theorem>[transverse] to the zero section, equivalently when $F$ and all of its homogeneous first partial derivatives have no common projective zero. Since
$$
K_X\cong\mathcal O(-n-1,-m-1),
$$
the <Adjunction formula> yields
$$
K_Y\cong\mathcal O(d_1-n-1,d_2-m-1)|_Y.
$$
Thus a smooth $F$ with $d_1=n+1$ and $d_2=m+1$ defines a <complex submanifold> with trivial <canonical bundle>.
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