## B-N-R Part 2: Moduli of Vector Bundles

Last time, we talked about twisted endomorphisms.  Now, we’re moving on to the second paragraph of the paper: generalized theta divisors.  In the meantime, we’re going to have to talk a bit about vector bundles and their moduli.

Let $X$ be a curve, and let $E$ be a vector bundle.  We define the slope of $E$ to be $\mu(E)=\frac{\deg(E)}{\mathrm{rank}(E)}$.  Now, we define a bundle to be (semi)stable if for every proper vector subbundle $E'\subset E$ we have $\mu(E')<\mu(E)$ ($\mu(E')\leq \mu(E)$).  We require stability in order to make much sense of the notion of a moduli space: it turns out to be the right notion to rigidify things, and is connected to the notion of stable and semistable points discussed in Geometric Invariant Theory.  We go to semistability in order to get compact moduli spaces.  One nice thing about the notion of semistability over stability is that it is characterized by a theorem of Faltings:

Thm: A vector bundle $E$ is semistable iff there exists a vector bundle $F$ such that $H^0(E\otimes F)=H^1(E\otimes F)=0$.

That’s it.  It’s semistable if you can tensor with something to kill the cohomology groups.  Even better, you don’t need to be over $\mathbb{C}$ for this to work, or even an algebraically closed field, though I don’t know the sharpest version out there.  Additionally, there are results that bound the rank of $F$ that you need to do this for semistable bundles.

You can use this theorem to prove that if $E$ is semistable, so is $L\otimes E$ for any line bundle $L$, and so is $E^\vee$, though these results can also be shown directly.  These two facts tell us some things about the moduli space of semistable vector bundles (hereby just referred to as the moduli of vector bundles).  First up, it gives us an involution.  We can take each bundle to its dual, and know that we still have a stable bundle.  Second, we get an action by the Jacobian of the curve $X$.  The only reason we restrict to the Jacobian rather than the whole Picard group is to keep the degree constant, if we’re looking at the space of all vector bundles of rank $n$, which would have countably many components, then we get an action of $\mathrm{Pic}(X)$.

So now, we take the moduli space of rank $n$ and degree $d$ vector bundles on $X$ and call it $\mathscr{U}_X(n,d)$.  The $U$ there refers to a connection to the unitary group, which we might talk about in the future, but won’t for the moment.  Now, for reasons that will be clear later, we want to have $\chi(E)=h^0(E)-h^1(E)=0$.  To get there, we use the Grothendieck-Riemann-Roch theorem, which states that for a vector bundle $E$, we have $\chi(E)=\deg E-\mathrm{rank}(E)(g-1)$.  So we get $0=d-n(g-1)$, and so we want to restrict to $\mathscr{U}_X(n,n(g-1))$.

Once we’ve got this moduli space, we want to talk about families of vector bundles.  Let $T$ be a variety and have it index a family $E_t$ of bundles on $X$, given by $E$ on $X\times T$.  Set $p:X\times T\to T$ to be the projection.  We are going to look at $R^1p_*(E)$.  Now, higher pushforwards are nice with respect to pullbacks, which we need because we’re really shooting at defining a universal object in the end.  For now, we’ll just look at the support of $R^1p_*(E)$.  This sheaf is the sheafification of the assignment $V\mapsto H^1(X\times V,E|_{X\times V})$, and so will be supported on $\{t\in T|H^1(E_t)\neq 0\}$.  Now, we have $h^0(E)-h^1(E)=0$, and so $H^1(E)\neq 0$, which is hard to check, is equivalent to $H^0(E)\neq 0$, which is better.  So $R^1p_*(E)$ is supported on $\{t\in T|H^0(E_t)\neq 0\}$.

Now, if you’ve done anything with Jacobians of curves and their Theta Divisors, you’ll recognize this.  There, you get a divisor consisting of those line bundles that have a nontrivial global section.  Here, we need to do a bit more work, because $R^1p_*(E)$ isn’t generally a line bundle.  So we look at $(\det R^1p_*(E))\otimes (\det p_*E)^{-1}$, and define it to be the line bundle $\mathscr{O}(\Theta_T)$.  This is a sort of relative version of a Theta Divisor.  The real thing we’re looking for, though, is on $\mathscr{U}_X(n,n(g-1))$.  What we get there is a line bundle $\mathscr{O}(\Theta)$, defined by the property that given a family indexed by $T$, we get a classifying map $f:T\to\mathscr{U}_X(n,n(g-1))$, and we require that $f^*\Theta=\Theta_T$.  Such a divisor exists, and is called a Generalized Theta Divisor on the Moduli Space of Vector Bundles.