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Symplectic geometry of the moduli space of projective structures in homological coordinates  [PDF]
Marco Bertola,Dmitry Korotkin,Chaya Norton
Mathematics , 2015,
Abstract: We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables. The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.
Symplectic and Poisson structures of certain moduli spaces  [PDF]
Johannes Huebschmann
Physics , 1993,
Abstract: Symplectic and Poisson structures of certain moduli spaces/Huebschmann,J./ Abstract: Let $\pi$ be the fundamental group of a closed surface and $G$ a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction due to A. Weinstein relying on techniques from equivariant cohomology may be refined so as to yield (i) a symplectic structure on a certain smooth manifold $\Cal M(\Cal P,G)$ containing the space $\roman{Hom}(\pi,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M(\Cal P,G)$ preserving the symplectic structure, with momentum mapping $\mu \colon \Cal M(\Cal P,G) \to g^*$, in such a way that the reduced space equals the space $\roman{Rep}(\pi,G)$ of representations. Our approach is somewhat more general in that it also applies to twisted moduli spaces; in particular, it yields the {\smc Narasimhan-Seshadri} moduli spaces of semistable holomorphic vector bundles by {\it symplectic reduction in finite dimensions}.This implies that, when the group $G$ is compact, such a twisted moduli space inherits a structure of {\it stratified symplectic space}, and that the strata of these twisted moduli spaces have finite symplectic volume.
On the symplectic structure over a moduli space of orbifold projective structures  [PDF]
Pablo Ares-Gastesi,Indranil Biswas
Mathematics , 2013,
Abstract: Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichm\"uller space for S. The total space of the holomorphic cotangent bundle of the Teichm\"uller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification is proved to be compatible with these symplectic structures. Similar results are obtained for biholomorphisms constructed using uniformizations provided by Schottky groups and Earle's version of simultaneous uniformization.
The Second Cohomology with Symplectic Coefficients of the Moduli Space of Smooth Projective Curves  [PDF]
Alexandre Kabanov
Mathematics , 1996,
Abstract: Each finite dimensional irreducible rational representation V of the symplectic group Sp_{2g} determines a generically defined local system \V over the moduli space M_g of genus g smooth projective curves. We study H^2(M_g;\V) and the mixed Hodge structure on it. Specifically, we prove that if g>5, then the natural map IH^2(MS_g;\V)-->H^2(M_g;\V) is an isomorphism where MS_g is the Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H^2(M_g;\V) is pure of weight 2+r if \V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H^2(M_g;\V) for 2
Moduli of symplectic instanton vector bundles of higher rank on projective space $\mathbb{P}^3$  [PDF]
Ugo Bruzzo,D. Markushevich,A. S. Tikhomirov
Mathematics , 2011,
Abstract: Symplectic instanton vector bundles on the projective space $\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\mathbb{P}^3$ with $r\ge2$ and second Chern class $n\ge r,\ n\equiv r({\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.
Singular symplectic moduli spaces  [PDF]
Dmitry Kaledin,Manfred Lehn,Christoph Sorger
Mathematics , 2005, DOI: 10.1007/s00222-005-0484-6
Abstract: Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O'Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension $\geq4$ they do no admit projective symplectic resolutions.
Automorphisms of moduli spaces of symplectic bundle  [PDF]
Indranil Biswas,Tomas L. Gomez,Vicente Mu?oz
Mathematics , 2010,
Abstract: Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE ---> L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_{Sp}(L) is generated by automorphisms sending E to ExM, where M is a 2-torsion line bundle, and automorphisms induced by automorphisms of X.
Lie Algebroids and generalized projective structures on Riemann surfaces  [PDF]
A. Levin,M. Olshanetsky
Mathematics , 2007,
Abstract: The space of generalized projective structures on a Riemann surface $\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on $\Sigma\times\mathbb{R}$. This space is a generalization of the space of projective structures on the Riemann surface. We define the moduli space of $W_N$-gravity as a symplectic quotient with respect to the canonical action of a special class of Lie algebroids. This moduli space describes in particular the moduli space of deformations of complex structures on the Riemann surface by differential operators of finite order, or equivalently, by a quotient space of Volterra operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids, because they are constructed by means of AGD bivector on the space of opers restricted on a circle. The AGD-algebroids are particular case of Lie algebroids related to a Poisson sigma-model. The moduli space of the generalized projective structure can be described by cohomology of a BRST-complex.
Minimal surfaces and symplectic structures of moduli spaces  [PDF]
Brice Loustau
Mathematics , 2014,
Abstract: Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.
Symplectic and Poisson geometry of the moduli spaces of flat connections over quilted surfaces  [PDF]
David Li-Bland,Pavol Severa
Mathematics , 2013,
Abstract: In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.
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