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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Choe, Insong, and George H. Hitching. "Non-defectivity of Grassmannian bundles over a curve." International Journal of Mathematics 27, no. 07 (June 2016): 1640002. http://dx.doi.org/10.1142/s0129167x16400024.

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Анотація:

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

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2

Choe, Insong, and G. H. Hitching. "A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve." International Journal of Mathematics 25, no. 05 (May 2014): 1450047. http://dx.doi.org/10.1142/s0129167x14500475.

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Анотація:

A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces. We give a sharp upper bound on t(V), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

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3

Benedetti, Gabriele, and Alexander F. Ritter. "Invariance of symplectic cohomology and twisted cotangent bundles over surfaces." International Journal of Mathematics 31, no. 09 (July 30, 2020): 2050070. http://dx.doi.org/10.1142/s0129167x20500706.

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Анотація:

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

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4

CHOE, INSONG, and GEORGE H. HITCHING. "Lagrangian subbundles of symplectic bundles over a curve." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 2 (February 22, 2012): 193–214. http://dx.doi.org/10.1017/s0305004112000096.

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Анотація:

AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

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5

Bakuradze, M. "On the Buchstaber Subring in MSp∗." gmj 5, no. 5 (October 1998): 401–14. http://dx.doi.org/10.1515/gmj.1998.401.

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Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

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6

HITCHING, GEORGE H. "RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO." International Journal of Mathematics 19, no. 04 (April 2008): 387–420. http://dx.doi.org/10.1142/s0129167x08004716.

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Анотація:

The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

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7

de Araujo, Artur. "Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences." International Journal of Mathematics 30, no. 03 (March 2019): 1850085. http://dx.doi.org/10.1142/s0129167x18500854.

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We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

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8

BISWAS, INDRANIL, та SARBESWAR PAL. "ON MODULI SPACE OF HIGGS Gp(2n, ℂ)-BUNDLES OVER A RIEMANN SURFACE". International Journal of Geometric Methods in Modern Physics 07, № 02 (березень 2010): 311–22. http://dx.doi.org/10.1142/s0219887810004002.

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Let X be a compact connected Riemann surface; the holomorphic cotangent bundle of X will be denoted by KX. Let [Formula: see text] denote the moduli space of semistable Higgs Gp (2n, ℂ)-bundles over X of fixed topological type. The complex variety [Formula: see text] has a natural holomorphic symplectic structure. On the other hand, for any ℓ ≥ 1, the Liouville symplectic from on the total space of KX defines a holomorphic symplectic structure on the Hilbert scheme Hilb ℓ(KX) parametrizing the zero-dimensional subschemes of KX. We relate the symplectic form on Hilb ℓ(KX) with the symplectic form on [Formula: see text].

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9

BISWAS, INDRANIL, TOMAS L. GÓMEZ, and VICENTE MUÑOZ. "AUTOMORPHISMS OF MODULI SPACES OF SYMPLECTIC BUNDLES." International Journal of Mathematics 23, no. 05 (May 2012): 1250052. http://dx.doi.org/10.1142/s0129167x12500528.

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Анотація:

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where [Formula: see text], together with the automorphisms induced by automorphisms of X.

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10

Otiman, Alexandra. "Locally conformally symplectic bundles." Journal of Symplectic Geometry 16, no. 5 (2018): 1377–408. http://dx.doi.org/10.4310/jsg.2018.v16.n5.a5.

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11

BISWAS, INDRANIL, and TOMÁS L. GÓMEZ. "HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES." International Journal of Mathematics 17, no. 01 (January 2006): 45–63. http://dx.doi.org/10.1142/s0129167x06003357.

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Анотація:

We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

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12

Ginzburg, Viktor L., and Jeongmin Shon. "On the filtered symplectic homology of prequantization bundles." International Journal of Mathematics 29, no. 11 (October 2018): 1850071. http://dx.doi.org/10.1142/s0129167x18500714.

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We study Reeb dynamics on prequantization circle bundles and the filtered (equivariant) symplectic homology of prequantization line bundles, aka negative line bundles, with symplectically aspherical base. We define (equivariant) symplectic capacities, obtain an upper bound on their growth, prove uniform instability of the filtered symplectic homology and touch upon the question of stable displacement. We also introduce a new algebraic structure on the positive (equivariant) symplectic homology capturing the free homotopy class of a closed Reeb orbit — the linking number filtration — and use it to give a new proof of the non-degenerate case of the contact Conley conjecture (i.e. the existence of infinitely many simple closed Reeb orbits), not relying on contact homology.

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13

Bunk, Severin. "Gerbes in Geometry, Field Theory, and Quantisation." Complex Manifolds 8, no. 1 (January 1, 2021): 150–82. http://dx.doi.org/10.1515/coma-2020-0112.

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Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

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14

Hsieh, Po-Hsun. "Symplectic Geometry of Vector Bundle Maps of Tangent Bundles." Rocky Mountain Journal of Mathematics 31, no. 3 (September 2001): 987–1001. http://dx.doi.org/10.1216/rmjm/1020171675.

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15

Lalonde, François, and Dusa McDuff. "Symplectic structures on fiber bundles." Topology 42, no. 2 (March 2003): 309–47. http://dx.doi.org/10.1016/s0040-9383(01)00020-9.

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16

Biswas, Indranil, Souradeep Majumder, and Michael Lennox Wong. "Orthogonal and symplectic parabolic bundles." Journal of Geometry and Physics 61, no. 8 (August 2011): 1462–75. http://dx.doi.org/10.1016/j.geomphys.2011.03.009.

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17

Ojanguren, M., R. Parimala, and R. Sridharan. "Symplectic bundles over affine surfaces." Commentarii Mathematici Helvetici 61, no. 1 (December 1986): 491–500. http://dx.doi.org/10.1007/bf02621929.

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18

Choe, Insong, and George H. Hitching. "Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve." International Journal of Mathematics 26, no. 13 (December 2015): 1550106. http://dx.doi.org/10.1142/s0129167x15501062.

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Анотація:

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions.

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19

CRASMAREANU, MIRCEA. "DIRAC STRUCTURES FROM LIE INTEGRABILITY." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1220005. http://dx.doi.org/10.1142/s0219887812200058.

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We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

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20

Schnitzer, Jonas, and Luca Vitagliano. "The Local Structure of Generalized Contact Bundles." International Mathematics Research Notices 2020, no. 20 (February 25, 2019): 6871–925. http://dx.doi.org/10.1093/imrn/rnz009.

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Abstract Generalized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

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21

Vinberg, E. "Equivariant Symplectic Geometry of Cotangent Bundles." Moscow Mathematical Journal 1, no. 2 (2001): 287–99. http://dx.doi.org/10.17323/1609-4514-2001-1-2-287-299.

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22

Ebner, Oliver, and Stefan Haller. "Harmonic cohomology of symplectic fiber bundles." Proceedings of the American Mathematical Society 139, no. 08 (August 1, 2011): 2927. http://dx.doi.org/10.1090/s0002-9939-2010-10707-4.

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23

Bowden, Jonathan. "The topology of symplectic circle bundles." Transactions of the American Mathematical Society 361, no. 10 (October 1, 2009): 5457. http://dx.doi.org/10.1090/s0002-9947-09-04721-7.

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24

Gotay, Mark J. "On symplectic submanifolds of cotangent bundles." Letters in Mathematical Physics 29, no. 4 (December 1993): 271–79. http://dx.doi.org/10.1007/bf00750961.

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25

Chantraine, Baptiste, and Emmy Murphy. "Conformal symplectic geometry of cotangent bundles." Journal of Symplectic Geometry 17, no. 3 (2019): 639–61. http://dx.doi.org/10.4310/jsg.2019.v17.n3.a2.

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26

Panin, I., and C. Walter. "Quaternionic Grassmannians and Borel classes in algebraic geometry." St. Petersburg Mathematical Journal 33, no. 1 (December 28, 2021): 97–140. http://dx.doi.org/10.1090/spmj/1692.

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Анотація:

The quaternionic Grassmannian H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) is the affine open subscheme of the usual Grassmannian parametrizing those 2 r 2r -dimensional subspaces of a 2 n 2n -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have HP n = H Gr ⁡ ( 1 , n + 1 ) \operatorname {HP}^n = \operatorname {H Gr}(1,n+1) . For a symplectically oriented cohomology theory A A , including oriented theories but also the Hermitian K \operatorname {K} -theory, Witt groups, and algebraic symplectic cobordism, we have A ( HP n ) = A ( pt ) [ p ] / ( p n + 1 ) A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1}) . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes. The cell structure of the H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is HP n \operatorname {HP}^n where the cell of codimension 2 i 2i is a quasi-affine quotient of A 4 n − 2 i + 1 \mathbb {A}^{4n-2i+1} by a nonlinear action of G a \mathbb {G}_a .

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27

VAISMAN, IZU. "COUPLING POISSON AND JACOBI STRUCTURES ON FOLIATED MANIFOLDS." International Journal of Geometric Methods in Modern Physics 01, no. 05 (October 2004): 607–37. http://dx.doi.org/10.1142/s0219887804000307.

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Let M be a differentiable manifold endowed with a foliation ℱ. A Poisson structure P on M is ℱ-coupling if ♯P(ann(Tℱ)) is a normal bundle of the foliation. This notion extends Sternberg's coupling symplectic form of a particle in a Yang–Mills field [11]. In the present paper we extend Vorobiev's theory of coupling Poisson structures [16] from fiber bundles to foliated manifolds and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. We then discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

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28

Kurek, J., and W. M. Mikulski. "Symplectic structures on the tangent bundles of symplectic and cosymplectic manifolds." Annales Polonici Mathematici 82, no. 3 (2003): 273–85. http://dx.doi.org/10.4064/ap82-3-8.

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29

Bates, Larry M. "Examples for obstructions to action-angle coordinates." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 1-2 (1988): 27–30. http://dx.doi.org/10.1017/s0308210500024823.

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SynopsisWe give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.

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30

Tikhomirov, A. S., S. A. Tikhomirov, and D. A. Vasiliev. "Construction of stable rank 2 bundles on P3 via symplectic bundles." Sibirskii matematicheskii zhurnal 60, no. 2 (December 4, 2018): 441–60. http://dx.doi.org/10.33048/smzh.2019.60.215.

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31

Tikhomirov, A. S., S. A. Tikhomirov та D. A. Vassiliev. "Construction of Stable Rank 2 Bundles on ℙ3 Via Symplectic Bundles". Siberian Mathematical Journal 60, № 2 (березень 2019): 343–58. http://dx.doi.org/10.1134/s0037446619020150.

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32

BISWAS, INDRANIL, and AVIJIT MUKHERJEE. "QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES." International Journal of Mathematics 15, no. 09 (November 2004): 907–17. http://dx.doi.org/10.1142/s0129167x04002594.

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Анотація:

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

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33

Kobayashi, Shoshichi. "Simple vector bundles over symplectic Kähler manifolds." Proceedings of the Japan Academy, Series A, Mathematical Sciences 62, no. 1 (1986): 21–24. http://dx.doi.org/10.3792/pjaa.62.21.

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34

Timashev, D. "Equivariant Symplectic Geometry of Cotangent Bundles, II." Moscow Mathematical Journal 6, no. 2 (2006): 389–404. http://dx.doi.org/10.17323/1609-4514-2006-6-2-389-404.

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35

Lacirasella, Ignazio, Juan Carlos Marrero, and Edith Padrón. "Reduction of symplectic principal $\mathbb R$-bundles." Journal of Physics A: Mathematical and Theoretical 45, no. 32 (July 24, 2012): 325202. http://dx.doi.org/10.1088/1751-8113/45/32/325202.

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36

Ding, Fan, and Youlin Li. "Strong symplectic fillability of contact torus bundles." Geometriae Dedicata 195, no. 1 (November 15, 2017): 403–15. http://dx.doi.org/10.1007/s10711-017-0299-9.

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37

Biswas, Indranil, and Tomás L. Gómez. "Stability of symplectic and orthogonal Poincaré bundles." Journal of Geometry and Physics 76 (February 2014): 97–106. http://dx.doi.org/10.1016/j.geomphys.2013.10.014.

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38

Maley, F. Miller, Jean Mastrangeli, and Lisa Traynor. "Symplectic Packings in Cotangent Bundles of Tori." Experimental Mathematics 9, no. 3 (January 2000): 435–55. http://dx.doi.org/10.1080/10586458.2000.10504420.

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39

Zinger, Aleksey. "Enumerative vs.\ symplectic invariants and obstruction bundles." Journal of Symplectic Geometry 2, no. 4 (2004): 445–543. http://dx.doi.org/10.4310/jsg.2004.v2.n4.a1.

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40

Gorbunov, Vassily, and Nigel Ray. "Orientations of Spin bundles and symplectic cobordism." Publications of the Research Institute for Mathematical Sciences 28, no. 1 (1992): 39–55. http://dx.doi.org/10.2977/prims/1195168855.

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41

Hurtubise, J. C., and L. C. Jeffrey. "Representations with Weighted Frames and Framed Parabolic Bundles." Canadian Journal of Mathematics 52, no. 6 (December 1, 2000): 1235–68. http://dx.doi.org/10.4153/cjm-2000-052-4.

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AbstractThere is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety Mh of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group G, with fixed conjugacy classes h at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For G = SU(2), we build a symplectic variety P of pairs (representations of the fundamental group into G, “weighted frame” at the puncture points), and a corresponding complex variety of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces Mh, , in the sense that one can obtain Mh from P by symplectic reduction, andMh from by a complex quotient. This allows us to explain certain features of the toric geometry of the SU(2) moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

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42

MARTÍNEZ TORRES, DAVID. "NONLINEAR SYMPLECTIC GRASSMANNIANS AND HAMILTONIAN ACTIONS IN PREQUANTUM LINE BUNDLES." International Journal of Geometric Methods in Modern Physics 09, no. 01 (February 2012): 1250001. http://dx.doi.org/10.1142/s0219887812500016.

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In this paper we extend to the Fréchet setting the following well-known fact about finite-dimensional symplectic geometry: if a Lie group G acts on a symplectic manifold in a Hamiltonian fashion with momentum map μ, given x ∈ M the isotropy group Gx acts linearly on the tangent space in a Hamiltonian fashion, with momentum map the Taylor expansion of μ up to degree 2. We use this result to give a conceptual explanation for a formula of Donaldson in [Scalar curvature and projective embeddings. I, J. Differential Geom.59(3) (2001) 479–522], which describes the momentum map of the Hamiltonian infinitesimal action of the Lie algebra of the group of Hamiltonian diffeomorphisms of a closed integral symplectic manifold, on sections of its prequantum line bundle.

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Moraru, Ruxandra. "Integrable Systems Associated to a Hopf Surface." Canadian Journal of Mathematics 55, no. 3 (June 1, 2003): 609–35. http://dx.doi.org/10.4153/cjm-2003-025-3.

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AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

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Baracco, Luca, and Giuseppe Zampieri. "Analytic discs in symplectic spaces." Nagoya Mathematical Journal 161 (March 2001): 55–67. http://dx.doi.org/10.1017/s0027763000022121.

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We develop some symplectic techniques to control the behavior under symplectic transformation of analytic discs A of X = ℂn tangent to a real generic submanifold R and contained in a wedge with edge R.We show that if A* is a lift of A to T* X and if χ is a symplectic transformation between neighborhoods of po and qo, then A is orthogonal to po if and only if Ã:= πχA* is orthogonal to qo. Also we give the (real) canonical form of the couples of hypersurfaces of ℝ2n ⋍ ℂn whose conormal bundles have clean intersection. This generalizes [10] to general dimension of intersection.Combining this result with the quantized action on sheaves of the “tuboidal” symplectic transformation, we show the following: If R, S are submanifolds of X with R ⊂ S and then the conditions can be characterized as opposite inclusions for the couple of closed half-spaces with conormal bundles In §3 we give some partial applications of the above result to the analytic hypoellipticity of CR hyperfunctions on higher codimensional manifolds by the aid of discs (cf. [2], [3] as for the case of hypersurfaces).

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Grady, Ryan, and Owen Gwilliam. "LIE ALGEBROIDS AS SPACES." Journal of the Institute of Mathematics of Jussieu 19, no. 2 (February 13, 2018): 487–535. http://dx.doi.org/10.1017/s1474748018000075.

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In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

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46

Kucharz, Wojciech. "Symplectic complex bundles over real algebraic four-folds." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 47, no. 3 (December 1989): 430–37. http://dx.doi.org/10.1017/s1446788700033152.

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AbstractLetXbe a compact affine real algebraic variety of dimension 4. We compute the Witt group of symplectic bilinear forms over the ring of regular functions fromXto C. The Witt group is expressed in terms of some subgroups of the cohomology groups.

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Kim, Hoil. "Moduli spaces of stable vector bundles on Enriques surfaces." Nagoya Mathematical Journal 150 (June 1998): 85–94. http://dx.doi.org/10.1017/s002776300002506x.

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Abstract.We show that the image of the moduli space of stable bundles on an Enriques surface by the pull back map is a Lagrangian subvariety in the moduli space of stable bundles, which is a symplectic variety, on the covering K3 surface. We also describe singularities and some other features of it.

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48

BHOSLE, USHA, INDRANIL BISWAS, and JACQUES HURTUBISE. "GRASSMANNIAN-FRAMED BUNDLES AND GENERALIZED PARABOLIC STRUCTURES." International Journal of Mathematics 24, no. 12 (November 2013): 1350090. http://dx.doi.org/10.1142/s0129167x13500900.

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We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin–Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

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Esen, Oğul, Hasan Gümral, and Serkan Sütlü. "Tulczyjew’s triplet for lie groups III: Higher order dynamics and reductions for iterated bundles." Theoretical and Applied Mechanics 48, no. 2 (2021): 201–36. http://dx.doi.org/10.2298/tam210312009e.

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Given a Lie group ??, we elaborate the dynamics on ??*??*?? and ??*????, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space ????*??, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics.

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50

Christian, Austin. "On symplectic fillings of virtually overtwisted torus bundles." Algebraic & Geometric Topology 21, no. 1 (February 25, 2021): 469–505. http://dx.doi.org/10.2140/agt.2021.21.469.

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