Unstable bundles on curves. "The maximally unstable ones correspond to Schwarzian differential equation on Riemann surfaces" ?

Question

What is known about unstable bundles on curves ? What is "maximally unstable bundle" and what means it corresponds to Schwarzian differential equation ?

Consider the P^1 with pairs of points glued p1=q1, p2=q2, ..., what can be said for such curves ?

Let me mention that for such curves bundles can be described just by the set of matrices M1, M2, .... which correspond to "gluing the fibers". This can be made precise algebraically as follows: functions on "glued curve" are function on P^1 which satisfy conditions f(p1)=f(q1), f(p2)=f(q2),... then modules can be described as vector functions on P1, such that s(p1)=M1 s(q1) , s(p2)=M2 s(q2) ,... clearly this is a module over the algebra above. It has been further studied in Serre's "Algebraic groups and class fields". Let me also mention our papers: http://arxiv.org/abs/hep-th/0303069 Hitchin system on singular curves I, http://arxiv.org/abs/hep-th/0309059 Hitchin systems on singular curves II. Gluing subschemes

PS Questions are inspired by discussion with Misha Kapovich after his answer on the following MO question What is the number field analogue of the Narasimhan-Seshadri theorem ?

Answer 1

Vector bundles on curves come equipped with a canonical filtration, the Harder Narasimhan filtration, defined (iteratively) maximal destabilizing subbundles. Maximally unstable just means that the HN filtration is a filtration by line bundles, ie is as refined as possible (so bundle is as far from stable as possible). There's a canonical example of such a bundle (up to the choice of spin structure/theta characteristic in the even rank case, if we're fixing the determinant to be trivial) which is an iterated extension of half-integral powers of the canonical bundle (with step of size one: there's a unique nontrivial extension of $\Omega^{k-1/2}$ by $\Omega^{k+1/2}$ up to isomorphism, represented by the fundamental class of the surface) -- these are called "indigenous bundles" in Gunning's books eg (or the work of S.Mochizuki). Connections on such a bundle are the same as so-called $sl_n$-opers - in the case of $n=2$, such connections are in bijection with projective structures on the Riemann surface, whose relation to Schwarzian derivatives (or the Virasoro central extension) is well documented in many places, eg Gunning again or the paper of Beilinson-Schechtman. This connection is elementary - you take the corresponding projective bundle, it carries a canonical section from the filtration and a connection which is everywhere transverse to this section, so integrating it defines a projective atlas on the curve.

Answer 2

Here are few more details for David's answer.

In the case of rank 2 holomorphic vector bundles over a compact Riemann surface $X$, a maximally unstable bundle is a bundle $V$ where the slope (in this case, the degree) of a destabilizing line subbundle $L$ in $V$ is maximal possible. In the case when $V$ is flat this just means that $L$ is a square root of the canonical bundle $K$ of $X$. The details were worked out by Robert Gunning in "Special Coordinate Coverings of Riemann Surfaces" Math. Annalen 170, p. 67 - 86 (1967). Faltings in "Real projective structures on Riemann surfaces", Compositio Math. 48 (1983), no. 2, 223–269, looked at the case of punctured Riemann surfaces. I did not see any detailed analysis in the case when $X$ is replaced by a singular stable curve, but it is probably not too hard.

To relate (in the compact case) maximally unstable bundles to monodromy, suppose that $\rho: \pi=\pi_1(X)\to PSL(2, {\mathbb C})$ is a representation equivariant with respect to a (locally injective) holomorphic function $f: \tilde{X}\to CP^1$ defined on the universal cover of $X$. One should think of $f$ as a holomorphic section (again denoted $f$) of the $CP^1$-bundle $P\to X$ associated to $\rho$. Local injectivity of $f$ translated to the fact that $f$, as a section, is transversal to the leaves of the flat connection on $P$.

A representation $\rho$ as above is known to lift (nonuniquely) to a representation $\tilde\rho: \pi\to SL(2, {\mathbb C})$. Let $V\to X$ be the associated (to $\tilde\rho$) flat rank 2 holomorphic vector bundle. The section $f$ lifts to a holomorphic line subbundle $L\subset V$. Then one does a computation and shows that $L$ is destabilizing and half-canonical, i.e., $L^2\cong K$. This procedure can be reversed, namely, a maximally destabilizing line subbundle in a rank 2 vector bundle projects to a section of a $CP^1$ bundle over $X$, etc.

To relate to Schwarzian, note that the function $f:\tilde{X}\to CP^1$ has a well-defined Schwarzian derivative $S(f)$ which projects to a holomorphic quadratic differential $q$ on $X$. (Holomorphicity of $q$ is equivalent to local conformality of $f$.) Then $\rho$ is the monodromy of the Schwarzian differential equation $S(f)=q$ where we now think of $f$ as a multivalued function on $X$. Conversely, given a holomorphic quadratic differential $q$, one recovers $f$ (uniquely up to postcomposition with elements of $PSL(2, {\mathbb C})$), representation $\rho$ etc.