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A Hermitian metric on a holomorphic vector bundle $E\to X$ is a smoothly varying family of positive-definite Hermitian forms $h_x$ on its fibers. A Chern connection is a connection on a vector bundle $\nabla$ that is compatible with $h$ and whose $(0,1)$ part is the bundle's Dolbeault partial connection, ${\nabla }^{0,1}={\bar{\partial }}_E$.

Choose a local holomorphic frame $e$ and write $h=h(e,e)>0$. The connection form in this frame isIf $e^{\prime }=ge$ for a nowhere-zero holomorphic function $g$, then $h^{\prime }=|g{|}^{2}h$ andwhich is exactly the connection-form transformation law. The local formulas therefore define a global connection.

Identify $\mathcal{O}(-1)$ with the tautological bundle over Complex projective space. The standard Hermitian inner product of ${\mathbb{C}}^{n+1}$ restricts to each tautological line. On the affine chart $U_i=\{Z_i\ne 0\}$, put $w_j=Z_j/Z_i$ for $j\ne i$ and use the holomorphic frameThenThe curvature form of a connection isConsequently Chern-Weil theory gives the closed representativewhere ${\omega }_{FS}$ is the Fubini-Study form.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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 $X$.

Solved by gpt-5.6-sol high.