Representations of surface groups via holomorphic connections - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:19:12Z http://mathoverflow.net/feeds/question/9006 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9006/representations-of-surface-groups-via-holomorphic-connections Representations of surface groups via holomorphic connections Joel Fine 2009-12-15T17:30:43Z 2009-12-15T22:36:18Z <p>EDIT: Tony Pantev has pointed out that the answer to this question will appear in forthcoming work of Bogomolov-Soloviev-Yotov. I look forward to reading it! </p> <h2>Background</h2> <p>Let $E \to X$ be a holomorphic vector bundle over a complex manifold. A connection $A$ in $E$ is called <i>holomorphic</i> if in local holomorphic trivialisations of $E$, $A$ is given by a holomorphic 1-form with values in End(E).</p> <p>Notice that the curvature of $A$ is necessarily a (2,0)-form. In particluar, holomorphic connections over Riemann surfaces are <i>flat</i>. This will be important for my question.</p> <h2>The Question</h2> <p>I am interested in the following situation. Let $E \to S$ be a rank 2 holomorphic vector bundle over a Riemann surface of genus $g \geq 2$. I suppose that $E$ admits a global holomorphic trivialisation (which I do <i>not</i> fix) and that we choose a nowhere vanishing section $v$ of $\Lambda^2 E$. (So I <i>do</i> fix a trivialisation of the determinant bundle.) I want to consider holomorphic connections in $E$ which make $v$ parallel. The holonomy of such a connection takes values in $\mathrm{SL}(2,\mathbb{C})$ (modulo conjugation). </p> <p>My question: if I allow you to change the complex structure on $S$, which conjugacy classes of representations of $\pi_1(S)$ in $\mathrm{SL}(2,\mathbb C)$ arise as the holonomy of such holomorphic connections?</p> <p>EDIT: As jvp points out, some reducible representations never arise this way. I actually had in mind irreducible representations, moreover with discrete image in $\mathrm{SL}(2,\mathbb{C})$. Sorry for not mentioning that in the beginning!</p> <h2>Motivation</h2> <p>A naive dimension count shows that in fact the two spaces have the same dimension: </p> <p>For the holomorphic connections, if you choose a holomorphic trivialisation of $E\to S$, then the connection is given by a holomorphic 1-form with values in $sl(2, \mathbb C)$. This is a $3g$ dimensional space. Changing the trivialisation corresponds to an action of $\mathrm{SL(2,\mathbb C)}$ and so there are in fact $3g-3$ inequivalent holomorphic connections for a fixed complex structure. Combined with the $3g -3$ dimensional space of complex structures on $S$ we see a moduli space of dimension $6g-6$.</p> <p>For the representations, the group $\pi_1(S)$ has a standard presentation with $2g$-generators and 1 relation. Hence the space of representations in $\mathrm{SL}(2,\mathbb{C})$ has dimension $6g-3$. Considering representations up to conjugation we subtract another 3 to arrive at the same number $6g-6$.</p> <h2>A curious remark</h2> <p>Notice that if we play this game with another group besides $\mathrm{SL}(2,\mathbb{C})$ which doesn't have dimension 3, then the two moduli spaces do not have the same dimension. So it seems that $\mathrm{SL}(2,\mathbb{C})$ should be important in the answer somehow.</p> http://mathoverflow.net/questions/9006/representations-of-surface-groups-via-holomorphic-connections/9009#9009 Answer by jvp for Representations of surface groups via holomorphic connections jvp 2009-12-15T18:01:01Z 2009-12-15T18:20:08Z <h2>An example</h2> <p>Consider any representation $\varrho$ of $\pi_1(S)$ into $\mathbb C^\ast$. The representation $\varrho \times \varrho^{-1}$ can be though as representation on $SL(2,\mathbb C)$. Any connection realizing this representation leave two line bundles invariant. These lines bundles are determined by the image of $\varrho$ and $\varrho^{-1}$ into $H^1(S, {\mathcal O^{\ast}}_S)$ by the natural morphism $$Hom(\pi_1(S), \mathbb C^{\ast}) \to H^1(S, \mathbb C^{\ast}) \to H^1(S,{\mathcal O_S}^{\ast})$$</p> <p>Thus in this case the representation determines the line-bundle, and it must be of the form $\mathcal L \oplus \mathcal L^*$. Of course the line-bundle $\mathcal L$ may be trivial for some complex structures but not for others. But if we start with a non-trivial representation with values in $S^1\subset \mathbb C^{\ast}$ then the line-bundle will not be trivial in not matter which complex structure since $H^1(S,S^1)$ is naturally isomorphic to $\ker H^1(S,\mathcal O_S^{\ast}) \to H^2(S, \mathbb Z)$.</p> <h2>Hilbert's 21st Problem</h2> <p>Your question is related to Hilbert's 21st problem. In it, instead of considering a compact Riemann surface of genus $g$ with a holomorphic connection on the trivial bundle one considers $\mathbb P^1$ minus a finite set $\Gamma$ of points with a meromorphic connection on the trivial bunle with at most simples poles on $\Gamma$. </p> <p>It is known that every representation of $\pi_1(\mathbb P^1 - \Gamma)$ on $SL(2,\mathbb C)$ is realized by a meromorphic connection on the trivial bundle with simple poles on $\Gamma$. I believe that this result is due to Birkhoff</p> <p>In Hilbert's 21st problem a parameter counting does not suffices to exclude the other groups $SL(n,\mathbb C)$, $n \ge 3$. Indeed Bolibruch proved that irreducible representations are always realizable, and constructed counter examples for the general case starting with dimension $n \ge 3$ if I remember correctly. Moreover, there are examples which show that the answer may depend on the analytic invariants of the set $\Gamma$.</p> http://mathoverflow.net/questions/9006/representations-of-surface-groups-via-holomorphic-connections/9030#9030 Answer by Jack Evans for Representations of surface groups via holomorphic connections Jack Evans 2009-12-15T19:21:14Z 2009-12-15T19:21:14Z <p>Have you compared this to Hitchin's 1987 paper "The Self Duality Equations on a Riemann Surface"? It's like the $SU(2)$ version of your $SL(2,\mathbb{R})$ one.</p> http://mathoverflow.net/questions/9006/representations-of-surface-groups-via-holomorphic-connections/9034#9034 Answer by Tony Pantev for Representations of surface groups via holomorphic connections Tony Pantev 2009-12-15T19:43:43Z 2009-12-15T20:16:02Z <p>This question is addressed in a very recent paper of Bogomolov-Soloviev-Yotov (I don't think it is on the web yet). Among many interesting things they prove that the map from the moduli space of pairs $(C,\nabla)$ where $\nabla$ is a holomorphic connection on the trivial rank two bundle on some smooth curve $C$ is submersive whenever $\nabla$ is irreducible and $C$ is generic. </p> <p>With regard to Jack Evans' comment: this is a very different question than the question of determining respresentations in a real form (which has been extensively studied by Hitchin, Goldman, Garcia-Prada, etc.). It is about a holomorphic subvariety in the moduli of representations. A better analogy will be to look at the moduli space of opers which is the moduli space of holomorphic flat connections on a fixed (non-trivial) rank two vector bundle, namely, the 1-st jet bundle of a theta characteristic on the curve. </p> http://mathoverflow.net/questions/9006/representations-of-surface-groups-via-holomorphic-connections/9049#9049 Answer by Jack Evans for Representations of surface groups via holomorphic connections Jack Evans 2009-12-15T22:36:18Z 2009-12-15T22:36:18Z <p>I'm too new to add this to my previous comment so apologies. </p> <p>Dmitri, the trivial bundle can be part of a stable Hitchin Pair (specifically if A and B are two matrices without common eigenspaces tensored with independent sections of the canonical bundle). </p> <p>The construction above generates maps from the Hitchin moduli space to itself if we start with the flat $SU(2)$ connections and use the Higgs field to define a holomorphic connection and then map the flat connection induced by the holomorphic connection to the one defined by the self-duality equations. Is this not likely to be holomorphic or well behaved under any of the complex structures? </p> <p>Likewise for a fixed $SU(2)$ representation there will be a map from the space constructed as suggested above to the Hitchin moduli space for any family of curves (or $SL(2,\mathbb{R})$ representation) and holomorphic connections.</p>