Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 17 / 2

A rank-$r$ holomorphic vector bundle is a complex manifold $E$ with a holomorphic projection to $X$ and local holomorphic trivializations $E{|}_{U_i}\cong U_i\times {\mathbb{C}}^r$ that are complex-linear on each fibre. Its transition maps have the form $(x,v)\mapsto (x,g_{ij}(x)v)$ with holomorphic $g_{ij}:U_i\cap U_j\to G{L}_r(\mathbb{C})$ satisfying the cocycle identities.

The holomorphic Picard group consists of isomorphism classes of holomorphic line bundles, with tensor product as multiplication, the trivial line bundle as identity and the dual bundle as inverse. The classification on the complex projective line givesThe inverse is degree; $\mathcal{O}(1)$ is the positive generator.

In a local trivialization, identify the lines in each fibre with $U_i\times {\mathbb{P}}^1$. On overlaps glue byThis action is well defined because $g_{ij}(x)$ is invertible and scaling $v$ does not change the resulting line. It is holomorphic: in affine projective coordinates it is a ratio of holomorphic functions on the open set where its denominator is nonzero. The transition maps satisfy the cocycle condition and have holomorphic inverses. They give the holomorphic projectivization by lines its complex-manifold atlas, of dimension ${\dim }_{\mathbb{C}}X+1$, on the usual projective-bundle topology. Points over different base points are separated by base neighborhoods; points over the same base point are separated inside a common local product chart. A countable trivializing cover and the standard charts of the complex projective line give second countability.

The local projections agree, defining a holomorphic map $p:\mathbb{P}(E)\to X$ whose fibre is $\mathbb{P}(E_x)\cong {\mathbb{P}}^1$.

Define the relative tautological line bundle $S\subset p^{\ast }E$ by $S_l=l$. Its local construction is holomorphic, and its restriction to a fibre is ${\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. The required relative hyperplane line bundle is thereforeThis fixes the lines convention for projectivization and the sign of the fibre degree.

Pullback and tensor product preserve holomorphic line bundles and their isomorphisms. Interpret negative tensor powers of $L$ as powers of $L^{\ast }$. Thus the formula defines a group homomorphism on isomorphism classes.

To prove injectivity, suppose $p^{\ast }M\otimes L^{\otimes n}$ is trivial. Restrict it to any fibre. A line pulled back from its base point is trivial there, so its fibre restriction is ${\mathcal{O}}_{{\mathbb{P}}^1}(n)$. By the Picard group classification, $n=0$. It remains to show that $p^{\ast }M$ trivial implies $M$ trivial.

Let $\sigma$ be a nowhere-zero holomorphic section of $p^{\ast }M$. On a sufficiently small open set where both $E$ and $M$ are trivial, write $\sigma =a_i(x,[v])p^{\ast }{e}_i$ for a frame $e_i$ of $M$. For each fixed $x$, $a_i(x,\cdot )$ is a holomorphic function on the compact complex projective line, hence is constant. It is therefore $b_i(x)$; evaluating at a constant projective point in the local trivialization shows $b_i$ is holomorphic in $x$. The section is nowhere zero, so each $b_i$ is nowhere zero. Its overlap laws are exactly those of a section of $M$, and $b_i{e}_i$ glue to a global holomorphic frame. Thus $M$ is trivial.

The kernel consists only of $({\mathcal{O}}_X,0)$, provingThis is the Picard injection for holomorphic projective bundles. The argument uses local fibrewise constancy, so no global section of the projective bundle is required.