Tautological vector bundle coming with a canonical inclusion

Question

The question arises from "Geometry of Algebraic Curves" from Arbarello, Cornalba, Griffiths and Harris. I wrote something down in my master's thesis that I admittedly don't completely understand, and I wanna figure it out now that I have some time on my hands!

Page 311 claims the following, which I expanded a bit to make the question self-contained :

---

Let $A$ and $B$ be algebraic vector bundles over a complex algebraic variety $W$ and let $\mathbb P \overset{def}= \mathbb P(\mathrm{Hom}(B,A)) \to W$ be the associated projective bundle. Let $\sigma : A \to B$ be a morphism of vector bundles ; by composition, $\sigma$ induces a map $$ \widetilde{\sigma} : \mathrm{Hom}(B,A) \to \mathrm{Hom}(A,A). $$ We consider the composition of bundle maps on $\mathbb P$ : $$ \mathcal O_{\mathbb P}(-1) \overset{i}{\to} \pi^* \mathrm{Hom}(B,A) \overset{\pi^* \widetilde{\sigma}}{\to} \pi^* \mathrm{Hom}(A,A) \overset{\mathrm{tr}}{\to} \mathcal O_{\mathbb P} $$ where $i$ is the standard inclusion and $\mathrm{tr}$ is the trace functional. (...)

---

What is this standard inclusion? I can't even deal with the case where $W$ is a point (which would at least let me understand the map on the fibers...). In this case vector bundles are vector spaces $V_A, V_B$, so I am looking for a map of varieties $\mathcal O_{\mathbb P(\mathrm{Hom}(V_B,V_A))}(-1) \to \pi_{\mathbb P}^* \mathrm{Hom}(V_B,V_A)$, and by the universal property of the pullback, since I already have a map $\mathcal O_{\mathbb P}(-1) \to \mathbb P$, all I need is a map $\mathcal O_{\mathbb P}(-1) \to \mathrm{Hom}(V_B,V_A)$. But even that map I don't see it!

Answer 1

It's pretty much the same as what happens for the tautological bundle over projective space.

Recall that if $\mathbb P^n$ is the projective space of lines in $V := \mathbb C^{n+1}$, the tautological bundle $\mathcal O(-1)$ is the bundle of lines in $V$, so we get an inclusion $$ 0 \to \mathcal O(-1) \to \underline V $$ of vector bundles over $\mathbb P^n$, where $\underline V$ is the trivial vector bundle over $\mathbb P^n$ with fiber $V$. The inclusion maps a line in $\mathcal O(-1)$ (which is the space of lines in $V$) to the line in $\underline V$, tautologically.

Passing now to your situation, suppose $E \to X$ is a holomorphic vector bundle of rank $r+1$ over a complex manifold $X$. We then construct the projective bundle $\pi : \mathbb P(E) \to X$ of lines in $E$, and get an inclusion $$ 0 \to \mathcal O(-1) \to \pi^* E $$ of vector bundles over $\mathbb P^n$, in exactly the same way as before. You have $E = \mathrm{Hom}(B,A)$, which doesn't really matter since the inclusion exists for any vector bundle.

Note that the first situation is just the case where $X$ is a point, because we can interpret a vector space $V$ as a vector bundle over a point, and then $\underline V = \pi^* V$, where $\pi : \mathbb P^n \to \{\mathrm{pt}\}$ is the constant map.

Answer 2

So, after a long discussion with Gunnar Þór Magnússon, I came to the following conclusion.

Algebraically, of course, the standard definitions are to begin with a complex variety $X$, a locally free sheaf of constant finite rank $\mathscr E$ on $X$ and then $\mathbb P(\mathscr E) \overset{def}= \mathrm{Proj}(\mathrm{Sym}(\mathscr E^*))$. The definitions of $\mathcal O_{\mathbb P \mathscr E}(k)$ follow from the $\mathrm{Proj}$ construction. In particular, if $X$ is a point, $\mathscr E$ corresponds to a vector space $V$ and $\mathcal O_{\mathbb P(V)}(k) = \widetilde{\mathrm{Sym}(V^*)(k)}$ for $k \in \mathbb Z$. In coordinates, this gives us $\mathcal O_{\mathbb P^n}(k) = \widetilde{ \mathbb C[x_0,\cdots,x_n](k) }$. However, this definition does not make it obvious how to define a map $\mathcal O_{\mathbb P V}(-1) \to \mathbb P(V) \times V$, showing that $\mathcal O_{\mathbb PV}(-1)$ is the tautological subbundle of the Grassmannian of lines in $V$, namely $\mathbb P(V) = \mathbb G(1,V)$, as described in this link. In the case of the tautological subbundle, I did understand the map, but I did not understand that the bundle was $\mathcal O_{\mathbb P(V)}(-1)$, so I ended up with equivalent problems but no understanding.

Gunnar gave me the link (see pages 277-281) describing the geometric description of $\mathcal O_{\mathbb P(V)}(1)$ via the dual of the exact sequence $$ 0 \to \mathcal O_{\mathbb P(V)}(-1) \to \underline V \to \underline V/\mathcal O_{\mathbb P(V)}(-1) \to 0 $$ to obtain $$ 0 \to (\underline V/\mathcal O_{\mathbb P(V)}(-1))^* \to \underline V^* \to \mathcal O_{\mathbb P(V)}(-1)^* \to 0 $$ We can use this to define geometrically $\mathcal O_{\mathbb P(V)}(1) \overset{def}= \mathcal O_{\mathbb P(V)}(-1)^*$. By "geometrically", I mean defining the classical variety set-theoretically (as in Hartshorne's first chapter or as in Harris' Algebraic Geometry, for instance) and then defining the map to $\mathbb P(V)$ making it into a geometric vector bundle. The above exact sequence shows that we can write $$ \mathcal O_{\mathbb P(V)}(-1)^* = \{ ([v],[\varphi : V \to \mathbb C]) \, | \, (*) \} $$ where $[v]$ is the line through $v \in V \setminus \{0\}$ and $\varphi_1,\varphi_2 : V \to \mathbb C$ are called equivalent at $[v]$ if $\varphi_1(v) = \varphi_2(v)$. When we fix an hyperplane $H \subseteq V$, we can identify the fiber of $\pi : \mathcal O_{\mathbb P(V)}(-1)^* \to \mathbb P(V)$ at $[v] \in \mathbb P(V) \setminus \mathbb P(H)$ with $V^*/H^*$ ; choosing a basis for $V$ also gives an identification $\mathcal O_{\mathbb P(V)}(-1)^* \simeq \widetilde{\mathrm{Sym}(V^*)(1)} = \mathcal O_{\mathbb P(V)}(1)$ which is consistent with changes of bases, so this isomorphism appears to be natural ; more explicitly, if $V = \langle e_0,\cdots,e_n \rangle$ and $V^* = \langle x_0,\cdots,x_n \rangle$ where $x_i(e_j) = \delta_{ij}$, since $\mathcal O_{\mathbb P(V)}(-1)^*|_{\mathcal D_+(x_i)}$ has fibers identified with $V^*/H_i^*$ (where $H_i$ is the hyperplane $x_i = 0$) and $\widetilde{\mathrm{Sym}(V^*)(1)}|_{\mathcal D_+(x_i)} \simeq \widetilde{\mathbb C[x_0/x_i,\cdots,x_n/x_i]}$, each quotient $\frac{x_j}{x_i}$ describes a section of both sheaves which can be used to identify them. I am not quite satisfied by this argument, but I am at least convinced. If someone has something better, I'm open.

The point is I am worried about naturality in $V$ (and more generally, in $\mathscr E$, but I guess it would follow from that in $V$) and choosing bases makes it not very believable. I want the functor which associates to each locally free sheaf of constant rank over $X$ a geometric vector bundle over $X$ to map $\widetilde{\mathrm{Sym}(V^*)}(k)$ to the variety $(\mathcal O_{\mathbb P(V)}(-1))(-k)$ where $\mathcal O_{\mathbb P(V)}(-1)$ is defined as the tautological line bundle which we then twist. I chose these two guys because they are the easiest to naturally define, but it might not be the way to go.

(I'd be a big fan of a coordinate free argument matching the algebraic and geometric definition of those sheaves.)