Holomorphic vector-bundle morphism - Imported Knowlpedia Content

Definition

Let $E\to X$ and $F\to Y$ be holomorphic vector bundles. A holomorphic vector-bundle morphism from $E$ to $F$ is a pair $(\Phi,f)$ in which $f:X\to Y$ and $\Phi:E\to F$ are holomorphic maps, the projections satisfy $\pi_F\circ\Phi=f\circ\pi_E$ , and each restriction $\Phi_x:E_x\to F_{f(x)}$ is complex-linear. Thus it is a vector-bundle morphism in the smooth category with the additional requirement of holomorphicity. A morphism over $X$ means $X=Y$ and $f=\operatorname{id}_X$ .

Local characterization

In holomorphic trivializations, $\Phi$ has the form

(x,v)\longmapsto\bigl(f(x),A(x)v\bigr),

where $A(x)$ is a matrix of holomorphic functions. Conversely, compatible matrices of holomorphic functions define such a morphism. The morphism is an isomorphism exactly when $f$ is a biholomorphism and every $A(x)$ is invertible.

Composition and induced maps

Holomorphic vector bundles and their morphisms form a category under composition. A morphism $E\to F$ over $X$ sends local holomorphic sections of $E$ to local holomorphic sections of $F$ . Duals, tensor products, direct sums, and exterior powers produce new holomorphic morphisms in the expected covariant or contravariant direction.

Rank and subbundles

If a morphism over $X$ has locally constant rank, its kernel and image are holomorphic vector subbundles, and its cokernel is a holomorphic vector bundle. Without constant rank, these objects generally belong to the category of coherent analytic sheaves rather than locally free bundles; fiberwise kernels alone do not guarantee local triviality Huybrechts, §2.2.

References

D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §2.2, morphisms and exact sequences of holomorphic vector bundles.