Transition functions for the tautological bundle

Define the tautological bundle over $CP^1$ to be $\tau = \{[a_1, a_2], (z_1, z_2) \in CP^1\times\mathbb{C}^2 | \exists \lambda \in \mathbb{C} \;\text{such that} \;\lambda (z_1,z_2) = (a_1, a_2) \}.$

Then $\tau$ trivializes over open sets $U_i = \{[a_1,a_2] | a_i \neq 0\}, i=1,2$ where $(\pi, \phi_i):\pi^{-1}(U_1) \rightarrow CP^1\times\mathbb{C}$ is given by $\phi_i([a_1,a_2]) = a_i.$ Note that $(\pi, \phi_1)^{-1}:U_1\times\mathbb{C} \rightarrow \pi^{-1}(U_1)$ is given by $([a_1,a_2], w) \rightarrow ([a_1, a_2], (w, \frac{a_2}{a_1}w)).$

Thus the transition function $T_{12}$ is given by $[a_1, a_2] \rightarrow \frac{a_2}{a_1}.$ Up to homotopy we need only look at the map on the equator of $CP^1 = S^2.$ Since we can identify $U_1$ with $S^2 \setminus \infty = \mathbb{C}$ by $[a_1,a_2] \rightarrow \frac{a_2}{a_1},$ the transition function $T_{12}$ appears to be a degree one map $S^1 \rightarrow S^1.$

Is this right? If so, why is this bundle always denoted $O_\mathbb{P}(-1)?$ Shouldn't it be called $O_\mathbb{P}(1)?$

Solution 1:

If $T_{12}$ has degree $1$ then $T_{21}$ has degree $-1$. At some stage you just have to make a decision about whether this bundle should be called $\mathcal{O}(1)$ or $\mathcal{O}(-1)$.

Here are three possible (related) reasons why the notation $\mathcal{O}(-1)$ is used for this bundle, or, rather, why $\mathcal{O}(1)$ is used for the bundle whose transition functions are the reciprocals of those for the tautological bundle. These reasons apply to $\mathbb{CP}^n$ for any $n$, not just to $\mathbb{CP}^1$, but I'll stick to the latter for simplicity.

Firstly, for $m \geq 0$ the bundle $\mathcal{O}(m)$ admits global holomorphic sections other than $0$. In fact the space of sections is isomorphic to the space of homogeneous degree $m$ polynomials in the homogeneous coordinates $a_1, a_2$. As an explicit example in the case $m=1$, the polynomial $\alpha_1 a_1 + \alpha_2 a_2$ corresponds to the function $\alpha_1 + \alpha_2 (a_2/a_1)$ in the trivialisation over $U_1$ and to $\alpha_1 (a_1/a_2) + \alpha_2$ in the trivialisation over $U_2$. This nice correspondence between sections of $\mathcal{O}(m)$ and degree $m$ polynomials is a good reason for the notation to be the way it is.

Secondly, for a collection of (not necessarily distinct) points $P_1, \ldots, P_m$ there is an associated line bundle $\mathcal{O}(P_1+\ldots + P_m)$, whose sections can be thought of as meromorphic functions on $\mathbb{CP}^1$ with at worst a simple pole at each $P_i$; if some of the $P_i$ coincide then higher order poles are allowed, corresponding to the multiplicity of the repeated point. (More generally for a divisor $D \subset \mathbb{CP}^n$ one can form a line bundle $\mathcal{O}(D)$.)

It turns out that the bundle $\mathcal{O}(P_1 + \ldots +P_m)$ only depends (up to isomorphism) on the value of $m$. Essentially this is because for any pair of distinct points $Q_1, Q_2 \in \mathbb{CP}^1$ there exists a meromorphic function with a simple pole at $Q_1$, a simple zero at $Q_2$, and no other zeros or poles: an example would be $(z-Q_2)/(z-Q_1)$. Anyway, if $P$ is your favourite point on $\mathbb{CP}^1$ then $\mathcal{O}(P_1 + \ldots + P_m) \cong \mathcal{O}(mP)$ and this bundle is isomorphic to the one we call $\mathcal{O}(m)$.

Of course you could now say, 'Why did we define $\mathcal{O}(P)$ the way we did; why did we not call that bundle $\mathcal{O}(-P)$ instead?' Well, for any complex line bundle $L$ on $\mathbb{CP}^1$ we can construct a class $c_1(L) \in H^2(\mathbb{CP}^1)$ called the first Chern class. This is a purely topological invariant, and is Poincare dual to the zero set of a generic smooth section of $L$. In the case of $\mathcal{O}(P)$, the point $P$ itself is the zero set of a section (which is transverse to the zero section, so is 'generic'), so is dual to the class $c_1(\mathcal{O}(P))$. This is nicer than $c_1(\mathcal{O}(P))$ being dual to $-P$.

Thirdly, there is a notion of positivity for line bundles that appears in results like the Kodaira vanishing and embedding theorems. Usually this is formulated in terms of positive curvature of metrics (which is related the class $c_1(L)$ in reason 2), and a typical consequence is the existence of 'lots' of holomorphic sections (tying in with reason 1). Happily the bundle $\mathcal{O}(m)$ on $\mathbb{CP}^n$ is positive if and only if $m > 0$.