Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 118 / 1

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 ${\mathbb{CP}}^2$.

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The Adjunction formula, $K_X\cong (K_{{\mathbb{CP}}^2}\otimes \mathcal{O}(d)){|}_X$, and $K_{{\mathbb{CP}}^2}\cong \mathcal{O}(-3)$ giveIf $h=c_1(\mathcal{O}(1))$, thenWith the standard integral normalization of the Fubini-Study form, this becomes $c_1(TX)=(3-d)[{\omega }_{FS}{|}_X/(2\pi )]$ in real cohomology.

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A Kähler manifold has a real closed $(1,1)$-form $\omega$ for which $g(u,v)=\omega (u,Jv)$ is positive definite. The Fubini-Study form ${\omega }_{FS}$ has these properties on ${\mathbb{CP}}^2$. Pullback by the holomorphic inclusion $i:X\hookrightarrow {\mathbb{CP}}^2$ preserves reality, type, and closedness, while positivity restricts to every nonzero vector in $TX$. Hence ${\omega }_X=i^{\ast }{\omega }_{FS}$ is a Kähler form on $X$.

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The curvature of the Chern connection on the complex line bundle $TX$ has type $(1,1)$, and $i{F}_{\nabla }$ is real. Since $X$ has complex dimension one, every real two-form is a unique smooth multiple of its nonvanishing area form, sofor some real smooth function $\lambda$. By Chern-Weil theory,The hyperplane class has degree $d$ on a degree-$d$ plane curve, so

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The form $i{F}_{\nabla }$ is the Ricci form of a Kähler manifold. In complex dimension one, $i{F}_{\nabla }=K_g{\omega }_X=\frac{1}{2}{\mathrm{Scal}}_g{\omega }_X$. Thuswhere $K_g$ is the Gaussian curvature.

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