Tags: #projects/research #projects/active # 2021-08-06 ![](attachments/20210704015029%201.png) - Associate a Hermitian form $\omega \da {i\over 2}(h - \bar{h}) \in \Omega^{1, 1}(X)$. - Can always get a canonical volume form $\vol_M \da \omega^n/n! \in \Omega^{n, n}(M)$, which in coordinates is $(i/2)^n \det h_{ab} dz^I \wedgeprod \bar{dz}^I$. - $\omega$ may or may not be closed, i.e. $d\omega = 0$. If so, $\omega$ is Kähler (and incidentally symplectic). ## 12:37 > Some notes on the following paper: - Common symplectic forms arise as the imaginary part of a Kähler Hermitian metric. - Moduli spaces that admit Kähler structures, associated to a single Riemann surface $X$? - The Jacobi variety (moduli space of degree 0 line bundles, so $\Pic^0(C)$ for $C$ an algebraic curve) - Teichmüller space - Moduli of stable vector bundles - $H^1(X; \RR)$? - Complex structures of these over-spaces may vary as the complex structure of $X$ is deformed, but the symplectic structure is a topological invariant. - **Goal**: unify all of these examples, interpret the resulting symplectic structure as arising from an intersection pairing on $X$. - $S$ a closed oriented topological surface, $\pi \da \pi_1(S)$, $G$ a connected Lie group, $\Hom(\pi, G)$ the real-analytic variety of representations in the compact-open topology - Note that if $G\in \Ab$ then \[ \Hom(\pi, G)/G = \Hom(\pi, G) = H^1(\Sigma; G)\in \Ab ,\] and is an algebraic variety if $G$ is an algebraic group. - $H\da \Hom(\pi, G) \in \mods{G}$ where $G$ acts by composing with $\Inn(G)$. - In generality, $\Hom(\pi, g) \in \bimod{\Aut \pi}{G}$. Since $G$ acts by $\Inn(G)$, we get $H\in \bimod{\Out \pi}{G}$. - Can take $H \da \Hom(\pi, G)/G$ -- supposed to look like "outer" representations? - If $G\in \Ab$ then $H \cong H^1(S; G) \in \Ab$. Under more general assumptions, $H$ admits a symplectic structure generalizing the Kähler forms on the spaces above. - **Main result**: The Weil-Petersson Kähler form on $\mct_S$ extends to a symplectic structure on $\Hom(\pi, G)/G$ for $G = \PSL_2(\RR)$ and $\PSL_2(\CC)$. - Also a new proof that the WP metric on $\mct$ is Kähler, using periods of quadratic differentials. - Fact: $\Out \pi \cong \pi_0 \Diff(S) = \MCG(S)$. $\mct_S \in \pi_0 H$ is a connected component, and $\mct_S / \Out \pi$ is the moduli of complex structures on $S$. - $H$ is singular: pick out simple points $H^-$ whose centralizers have minimal dimension, this is a possibly non-Hausdorff topological manifold? > Why this condition of centralizers with minimum dimension? The singularities are at worst quadratic. A symplectic structure is any closed nondegenerate 2-form $\omega$ on $H^-$, then try to extend $\omega$ to all of $H$. Obstructions may live in $\T H$, the Zariski tangent space? - *How do you define differential forms away from smooth manifolds..?* - There is a functor \[ \correspond{ (G, B), \text{ Lie groups with} \\ \text{a symmetric bilinar form on }\lieg } &\mapstofrom \correspond{ \text{Symplectic $G\dash$ spaces} \\ G \da \Out \pi } \\ (G, B) &\mapsto (\Hom(\pi,G)/G, \omega^{(B)} ) ,\] where $\omega^{(B)}$ is a symplectic structure defined by $B$. - For $G= U_n$, $H$ is the space of stable vector bundles. - Use Fox calculus to write $\omega^{(B)}$ as an algebraic tensor in the algebraic coordinates on $H$ as an explicit 2-cycle no a surface. - Can prove that the WP metric is Kähler by showing that the canonical Hermitian metric on the space of stable rank 2 vector bundles of Chern class zero is Kähler. - $\Hom(\pi, G)$ equipped with the compact-open topology: the paper says $\phi_n \to \phi$ iff $\phi_n(g) \to \phi(g)$ for all $g$. This sounds like pointwise convergence, but maybe this already implies uniform convergence of $\phi_n$ somehow..? Assuming $G$ is compact maybe? - Remarkable fact: points of $\Hom(\pi, G)$ where $G$ does not act locally freely are precisely the singular points. - Algebraic coordinates: take $G^{\times 2k}$, where $\pi = \gens{A_i, B_i}_{i\leq k}/\gens{\prod [A_i, B_i]}$, then the relation $\prod [A_i, B_i] = 1$ is polynomial in the variables $A_i, B_i$. So $H$ is an algebraic variety. - The Zariski tangent space: $Z^1(\pi; \lieg_{\ad_f})$ for $f\in H$, where $\lieg = \Lie(G)$. Found by taking a path $f_t$, identified with a crossed morphism $\pi \to \lieg_{\ad_f}$ in $\mods{\pi}$. Write $f_t(x) = e^{t u(x) + O(t^2) } f(x)$, enforcing $f_t(xy) = f_t(x) f_t(y)$ yields a cocycle condition \[ u(xy) = u(x) + \ad_{f(x)} u(y) .\] - Can expand to \[ f_t = \exp\qty{tu(x) + t^2 u_2(x) + O(t^3)} f(x) ,\] and finding a 2nd order term $u_2$ reduces to solving an equation enforcing a similar homomorphism condition: \[ u_{2}(x)-u_{2}(x y)+\operatorname{Ad} \phi(x) u_{2}(y)=\frac{1}{2}[u(x), \text { Ad } \phi(x) u(y)] .\] The first obstruction is the following product being nonzero: \[ [\xi, \xi]: H^p(\pi; \lieg_{\ad_f}) \tensor H^q(\pi; \lieg_{\ad_f}) &\to H^2(\pi; \lieg_{\ad_f}) \\ x\tensor y &\mapsto {1\over 2}[ u(x), \ad_{f(x)}u(y)]? .\] Get an infinite series of obstructions from coefficients in Taylor series for $f(x)\inv f_t(x)$, each taking values in $H^2(\pi; \lieg_{\ad_f})$. Here $\xi\in H^1(\pi, \lieg_{\ad_f})$ is tangent to a path. - Can use Poincare duality to compute dimensions: \[ \dim H^1(\pi; \lieg_{\ad_f}) = (2p-2) \dim G + 2\dim Z(f) .\] - Assume $G$ preserves a nondegenerate symmetric bilinear form $B$ on $\lieg$. Get a pairing \[ H^1(\pi; \lieg_{\ad_f}) \tensor H^1(\pi; \lieg_{\ad_f}) &\to H^2(\pi; \RR) \cong \RR ,\] defined by cup product on $\pi$ and $B$ as coefficient product. Regarding $H^1$ as $\T H$, regard this as a 2-tensor $\omega^{(B)}$. Showing it is closed takes some work. - Idea: view $H^1$ as de Rham cohomology of $S$ with coefficients in a flat vector bundle. Take complex $A^*(S; V)$ as $V\dash$valued differential forms on $S$. These are maybe sections of the form $\globsec{\Omega^*_X \tensor V}$, so locally $\omega \tensor s$ where $\omega$ is an $n\dash$form and $s$ is a section of $V$. - de Rham theorem: $H^*(A^\bullet(S; V)) = H_\sing^*(S; V) = H_{\Grp}^*(\pi; V)$ with $S = K(\pi, 1)$. - Go to a larger space $\liea$ of all connections on a certain principal $G\dash$bundle: ?