Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3

Let ${\mathcal{K}}_X^{\ast }$ be the sheaf of nonzero meromorphic functions. A local equation for a divisor on a complex manifold is determined modulo a nowhere-zero holomorphic factor and hence defines a section of the divisor sheaf on a complex manifold ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$. Conversely, local representatives of such a section have quotients in ${\mathcal{O}}_X^{\ast }$, so their zero and pole orders agree on overlaps and define a divisor. The constructions are inverse:

The exact sequenceinduces a long exact sequence in sheaf cohomology. Under $H^0({\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast })=\mathrm{Div}(X)$ and $H^1({\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$, its connecting homomorphism is the divisor-to-Picard map $D\mapsto \mathcal{O}(D)$. Exactness identifies its kernel with divisors of global nonzero meromorphic functions, namely principal divisors:

A holomorphic line bundle is ample when some positive tensor power is very ample, so its sections define an embedding into Complex projective space. It is a positive holomorphic line bundle when it has a Hermitian metric with positive Chern curvature $i{F}_{\nabla }$. The Kodaira embedding theorem says that a compact complex manifold carrying a positive holomorphic line bundle is projective; sufficiently high tensor powers give a holomorphic embedding into projective space.

Choose a very ample line bundle $A$ on $X$. By the theorem that a high ample twist is very ample, for sufficiently large $m$ bothare very ample. The dual bundle cancels the power of $A$, giving

Write $L\cong H_1\otimes H_2^{\ast }$ as in part (d). Each very ample line bundle is the pullback of $\mathcal{O}(1)$ under a projective embedding. A hyperplane section therefore supplies an effective divisor on a complex manifold $D_i$ with $H_i\cong \mathcal{O}(D_i)$. HenceEvery Picard group class is therefore in the image of the divisor-to-Picard map: