What is the meaning of the notation ${\cal O}_{\mathbb{CP}^n}(k)$?

Question

I am studying some papers in which the notation ${\cal O}_{\mathbb{CP}^1}(-1)$, ${\cal O}_{\mathbb{CP}^1}(-2)$ and ${\cal O}_{\mathbb{CP}^1}(1)$ appear. I am not familiar with that notation and while I have already found out ${\cal O}_{\mathbb{CP}^1}(-1)$ is the tautological line bundle, I haven't found what the others mean and want to understand this more generally, because surely there has to be some uniform definition that recovers the tautological line bundle as a special case.

I believe this is not special to $\mathbb{CP}^1$, and I also don't think there is something special about those numbers $-1,-2$ and $1$. So I imagine there is some general meaning for $\mathcal{O}_{\mathbb{CP}^n}(k)$ where $k\in \mathbb{Z}$.

My question is: what does $\mathcal{O}_{\mathbb{CP}^n}(k)$ mean and how to understand it? Is it really something defined for any $k\in \mathbb{Z}$ or for only some values like $-2,-1$ and $1$? If only for some numbers, why is it the case?

Answer 1

$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Sheaf}[1]{\mathcal{O}_{\mathbf{P}^{1}}(#1)}$tl; dr: These bundles are defined for each integer, and over projective spaces of arbitrary dimension. Geometrically, over the projective line:

• The total space of $\Sheaf{-1}$ may be identified with the blow-up of $\Cpx^{2}$ at the origin. The fibres are lines through the origin, and the exceptional curve is the zero section.
• The total space of $\Sheaf{1}$ may be identified with a punctured complex projective plane. The fibres are lines through the puncture, an arbitrary line not through the puncture may be taken as the zero section, and projection away from the puncture is the bundle map.
• $\Sheaf{-2}$ is the cotangent (i.e., canonical) bundle. The total space may be viewed as the quotient of $\Cpx^{2}$ under scalar multiplication by $-1$, blown up at the origin.
• $\Sheaf{2}$ is the tangent (anticanonical) bundle. Its unit circle bundle may be identified more-or-less naturally with the matrix group $SO(3)$. (The first column of a real $3 \times 3$ special-orthogonal matrix may be viewed as a point of the unit sphere, indentified with the Riemann sphere; the second column is orthogonal to the first, hence an arbitrary unit tangent vector to the sphere in real Cartesian three-space; the third column is uniquely determined by the first two.)

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In intrinsic detail, construct the projective line from two copies $U_{0}$ and $U_{1}$ of the complex line with coordinates $z^{0}$ and $z^{1}$. (It may be clearer at first to write $z = z^{0}$ and $w = z^{1}$. The indexing below generalizes to higher-dimensional projective spaces.) For each integer $k$, the holomorphic line bundle $\Sheaf{k}$ may be trivialized over $U_{0}$ and $U_{1}$ by coordinates $(z^{0}, \zeta^{0})$ and $(z^{1}, \zeta^{1})$, glued together by the identification $$ z^{0} = \frac{1}{z^{1}},\qquad \zeta^{0} = \frac{\zeta^{1}}{(z^{1})^k}. $$ That is, the transition function from $z^{0}$ to $z^{1}$ is $$ g_{01}(z^{0}) = \frac{1}{(z^{0})^{k}}. $$ A holomorphic section of $\Sheaf{k}$ is represented by holomorphic functions $\zeta^{0} = f_{0}(z^{0})$ and $\zeta^{1} = f_{1}(z^{1})$ satisfying $$ g_{01}(z^{0}) f_{0}(z^{0}) = f_{1}(z^{1}). $$