Complex manifolds, holomorphic vector bundles and Hermitian metrics

Let $m\ge 1$. A chart on a topological space $X$ is a homeomorphism $\phi :U\to \phi (U)$ from an open set $U\subseteq X$ onto an open subset of ${\mathbb{C}}^m$ with its usual topology (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); two charts $\phi ,\psi$ are holomorphically compatible when the transition maps $\psi \circ {\phi }^{-1}$ and $\phi \circ {\psi }^{-1}$ are holomorphic on the open sets where they are defined (Holomorphic functions on an open subset of ${\mathbb{C}}^m$). A holomorphic atlas is a family of pairwise compatible charts whose domains cover $X$. A complex manifold of dimension $m$ is a nonempty Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) second-countable (Second countability: an at most countable basis for the topology) topological space carrying a holomorphic atlas; the same maximal-atlas argument as for Riemann surfaces and holomorphic atlases applies verbatim with ${\mathbb{C}}^m$ in place of $\mathbb{C}$: the union of a holomorphic atlas with all charts compatible with it is again a holomorphic atlas, so one passes to the unique maximal atlas and calls its members holomorphic charts. Each chart is a homeomorphism onto an open subset of ${\mathbb{C}}^m\cong {\mathbb{R}}^{2m}$, so $X$ is a topological manifold without boundary of real dimension $2m$ (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces), and for connected $X$ the case $m=1$ recovers Riemann surfaces (Riemann surfaces and holomorphic atlases).

A holomorphic map $f:X\to Y$ between complex manifolds is a continuous map whose expression in every pair of holomorphic charts is holomorphic (Holomorphic maps ${\mathbb{C}}^m\to {\mathbb{C}}^n$ and the complex Jacobian matrix); equivalently, for every open $V\subseteq Y$ and every holomorphic function $h$ on $V$, the function $h\circ f$ is holomorphic on $f^{-1}(V)$. The forward implication is the holomorphic chain rule; conversely apply this condition to the coordinate functions of a chart on $Y$.

Induced smooth structure and complex structure. Holomorphic functions on open subsets of ${\mathbb{C}}^m$ are smooth (Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic), so every holomorphic atlas is a smooth atlas (Smooth atlases) and $X$ is a smooth manifold of real dimension $2m$ (Smooth manifolds and their smooth charts) whose smooth charts include the holomorphic ones. In holomorphic coordinates $z_k=x_k+i{y}_k$ the complex structure is the endomorphism field $J$ of the tangent bundle determined by $J{\mathrm{\partial }}_{x_k}={\mathrm{\partial }}_{y_k}$ and $J{\mathrm{\partial }}_{y_k}=-{\mathrm{\partial }}_{x_k}$; it satisfies $J^2=-\mathrm{i}\mathrm{d}$ and is independent of the holomorphic chart, because a holomorphic transition map has complex-linear differential (Holomorphic maps ${\mathbb{C}}^m\to {\mathbb{C}}^n$ and the complex Jacobian matrix, The composite of holomorphic maps is holomorphic and its complex Jacobian is the product) and complex-linear maps intertwine the two coordinate definitions of multiplication by $i$. A Hermitian metric on $X$ is a Riemannian metric $g$ (Riemannian metric and riemannian manifold) satisfying $g(Jv,Jw)=g(v,w)$ for all tangent vectors $v,w$; its Riemannian volume density is written ${\mu }_g$ (Riemannian volume density). The complex structure makes each tangent space a complex vector space, so the tangent bundle is a smooth complex vector bundle in the sense of Complex-linear and metric-compatible bundle connections.

Holomorphic vector bundles and Hermitian bundle metrics. A holomorphic vector bundle of rank $r$ on $X$ is a smooth complex vector bundle $\pi :E\to X$ of rank $r$ (Smooth vector bundles, rank, fibres, and trivial bundles, Complex-linear and metric-compatible bundle connections) together with an open cover $(V_{\alpha })$ of $X$ and smooth local trivializations ${\theta }_{\alpha }:E{\mathrm{\mid }}_{V_{\alpha }}\to V_{\alpha }\times {\mathbb{C}}^r$ whose transition matrices $g_{\alpha \beta }:V_{\alpha }\cap V_{\beta }\to {\mathrm{G}\mathrm{L}}_r(\mathbb{C})$, defined by ${\theta }_{\alpha }{\theta }_{\beta }^{-1}(p,v)=(p,g_{\alpha \beta }(p)v)$, are holomorphic. A holomorphic frame over an open set $V$ is a smooth complex-linear trivialization whose transition matrices with the supplied trivializations are holomorphic on every overlap. Changing holomorphic frame multiplies a local section by a holomorphic matrix, hence a smooth local section is holomorphic exactly when its representatives in holomorphic frames are holomorphic functions. A Hermitian metric $h$ on $E$ is a smooth bundle metric (Smooth bundle metrics) whose fibre pairings are positive definite Hermitian forms, linear in the first variable and conjugate-linear in the second; the condition is frame-independent because the gauge freedom is conjugation by a pointwise invertible matrix. Smooth sections are $\mathrm{\Gamma }(E)$ (Smooth sections, local sections, and support).

• Choice. The structures are supplied data. The maximal-atlas and gluing verifications above are local, and no existence of metrics or partitions is asserted; no choice principle is used in this definition.
• Conventions. The Hermitian convention is linear in the first variable, as in Smooth bundle metrics; the Riemannian metric $g$ is used on real vectors, and $J$ is declared from the holomorphic charts rather than assumed. The case $m=1$ reduces to Riemann surfaces and holomorphic atlases; the general-$m$ statements here are the ones used by the Dolbeault and elliptic items of this page.