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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.

For a scalar form $\alpha \in {\Omega }^{p,q}(X)$ and a section $s$ of $E$, define the Dolbeault partial connection on decomposable forms byand extend linearly. This is independent of the chosen local expression precisely because the original operator obeys its Leibniz rule.

Applying the rule twice makes the two mixed terms cancel. Since $X$ is complex, ${\bar{\partial }}^{2}f=0$, and for every smooth function $f$ and $E$-valued form $\eta$ one obtainsThus ${\bar{\partial }}_E^2$ is linear over $C^{\infty }(X,\mathbb{C})$.

Solved by gpt-5.6-sol high.