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For a complex submanifold $Y\subseteq X$, the holomorphic normal bundle is the quotient holomorphic vector bundle

The restriction of $J$ to the $J$-invariant bundle $TY$ is an 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 isIf $Y$ is a smooth hypersurface, taking top exterior powers in the holomorphic conormal sequencegivesThe normal bundle of a hypersurface is $N_{Y/X}\cong {\mathcal{O}}_X(Y){|}_Y$, so the Adjunction formula is

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 to the zero section, equivalently when $F$ and all of its homogeneous first partial derivatives have no common projective zero. Sincethe Adjunction formula yieldsThus a smooth $F$ with $d_1=n+1$ and $d_2=m+1$ defines a complex submanifold with trivial canonical bundle.

Solved by gpt-5.6-sol high.

On the affine chart $Z_i\ne 0$, divide the homogeneous polynomial $F$ by $Z_i^d$ to obtain a holomorphic function $f_i$ of two affine coordinates. The hypothesis $\mathrm{grad}F(p)\ne 0$ says that at every zero at least one affine partial derivative of $f_i$ is nonzero; the radial derivative contributes nothing on $F=0$ by Euler's homogeneous identity. The holomorphic implicit-function theorem therefore makes $f_i^{-1}(0)$ a complex submanifold of complex codimension one. These local loci agree on chart overlaps, so $X$ is a complex one-dimensional submanifold of ${\mathbb{CP}}^2$.

Solved by gpt-5.6-sol high.