Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 118 / 3 / Solution

A real (1, 1)-form is a form $\phi \in {\Omega }^{1,1}(X)$ satisfying $\bar{\phi }=\phi$. In holomorphic coordinates it has the formIt is a positive real (1, 1)-form whenfor every nonzero tangent vector $\xi$ of type $(1,0)$, equivalently when the Hermitian matrix $(h_{j\bar{k}})$ is positive definite.

A holomorphic local trivialization of a holomorphic line bundle $L\to X$ is equivalently a nowhere-zero holomorphic local frame $e$. A connection $A$ is unitary when it preserves the fiberwise Hermitian inner product:The Chern connection is the unique unitary connection whose $(0,1)$ part is the bundle's Dolbeault partial connection ${\bar{\partial }}_L$.

In a holomorphic frame, put $h=h(e,e)$. The local formula for the Chern connection on a line bundle isThe curvature therefore has type $(1,1)$. Since a unitary connection has imaginary curvature, $\overline{F(A)}=-F(A)$, and henceThus $iF(A)$ is a real $(1,1)$-form.

For connections $A$ on $L$ and $\hat{A}$ on $\hat{L}$, the tensor product connection is defined on decomposable local sections byThe curvature of a tensor product connection on line bundles is additive:Consequentlywhich is positive whenever both summands are positive.

For the final assertion, simultaneously diagonalize the positive Hermitian matrices of $\phi$ and $\psi$ by congruence at the chosen point. In the resulting coframe,A direct wedge-product calculation givesIf $\xi$ and $\eta$ are linearly independent, at least one of these $2\times 2$ minors is nonzero. Every coefficient is positive, so the sum is strictly positive. This is wedge positivity for two positive (1, 1)-forms.