Hermitian metric on a holomorphic vector bundle ($h$)

A Hermitian metric on a holomorphic vector bundle $E\to X$ is a smoothly varying positive-definite Hermitian form $h_x$ on every complex fiber $E_x$.

Such a metric makes the underlying smooth bundle a Hermitian vector bundle.

A holomorphic line bundle is semipositive if it admits a smooth Hermitian metric whose normalized Chern curvature form is semipositive. Tensoring a positive holomorphic line bundle with a nonnegative integer power of a semipositive one remains positive, because their curvature forms add. This smooth-metric property is stronger than simply having a nef numerical class.

With a Hermitian form linear in its first slot, $H(s\otimes \bar{t})=h(s,t)$ is a complex-linear dual tensor. The original connection, its conjugate connection, the tensor product connection and the dual connection induce $D_0$. Its derivative is $(D_{0,V}H)(s\otimes \bar{t})=Vh(s,t)-h(D_{V}s,t)-h(s,D_{V}t)$. Consequently $D_{0}H=0$ is exactly metric compatibility.

For a smooth subbundle $S$ of a Hermitian vector bundle $E$, orthogonal projection identifies $E/S$ smoothly with $S^{\perp }$. The quotient metric is $h_Q([v],[w])=h_E((I-{\pi }_S)v,(I-{\pi }_S)w)$. It is independent of representatives and positive definite. For a holomorphic subbundle the resulting smooth quotient metric is defined on the holomorphic quotient, but the orthogonal identification itself need not be holomorphic.

A holomorphic line bundle is positive when it admits a Hermitian metric whose Chern connection has positive curvature form $i{F}_{\nabla }$, equivalently when $i{F}_{\nabla }$ is a positive real (1, 1)-form.

The positive-curvature definition uses a positive real (1, 1)-form.

On a compact Kähler manifold $X$, if $P$ is a positive holomorphic line bundle, then $H^i(X,K_X\otimes P)=0$ for $i>0$. The canonical-bundle twist is part of the hypothesis and conclusion; positivity alone does not remove it on a general manifold. For a complex torus, the canonical bundle is trivial, so higher cohomology of a positive line bundle vanishes without an additional twist.

Let $L$ be positive and $V$ a closed smooth hypersurface in a complex torus. The semipositive canonical bundle of a submanifold of a complex torus makes $P_a=(L\otimes \mathcal{O}(aV)){|}_V$ positive. Since $K_V=\mathcal{O}(V){|}_V$, the restriction of the $a$-th ambient twist is $K_V\otimes P_{a-1}$ when $a\ge 1$. Apply the Kodaira vanishing theorem on $V$, and use $0\to L((a-1)V)\to L(aV)\to L(aV){|}_V\to 0$ and its long exact sequence to induct from the positive bundle $L$ on the torus. No unproved ambient positivity of $\mathcal{O}(V)$ is needed.

For a positive Hermitian holomorphic line bundle on compact $X$, using its curvature as the Kähler form, a linear lower bound in $k$ for the full Dolbeault Laplacian on positive antiholomorphic degrees rules out harmonic forms for sufficiently large $k$. The Dolbeault theorem and Dolbeault Hodge decomposition then imply $H^{q>0}(X,L^k)=0$.

The Kodaira embedding theorem says that a compact complex manifold carrying a positive holomorphic line bundle is projective: sufficiently large tensor powers of the bundle have global holomorphic sections defining a holomorphic embedding into Complex projective space. Conversely, the restriction of $\mathcal{O}(1)$ gives a positive line bundle on every projective complex manifold.