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Journal articles on the topic 'Symplectic manifolds Hodge theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

PAK, HONG KYUNG. "TRANSVERSAL HARMONIC THEORY FOR TRANSVERSALLY SYMPLECTIC FLOWS." Journal of the Australian Mathematical Society 84, no. 2 (April 2008): 233–45. http://dx.doi.org/10.1017/s1446788708000190.

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AbstractWe develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

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2

Allday, Christopher. "Canonical equivariant extensions using classical Hodge theory." International Journal of Mathematics and Mathematical Sciences 2005, no. 8 (2005): 1277–82. http://dx.doi.org/10.1155/ijmms.2005.1277.

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Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.

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3

Tseng, Li-Sheng, and Shing-Tung Yau. "Cohomology and Hodge Theory on Symplectic Manifolds: I." Journal of Differential Geometry 91, no. 3 (July 2012): 383–416. http://dx.doi.org/10.4310/jdg/1349292670.

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4

Tseng, Li-Sheng, and Shing-Tung Yau. "Cohomology and Hodge Theory on Symplectic Manifolds: II." Journal of Differential Geometry 91, no. 3 (July 2012): 417–43. http://dx.doi.org/10.4310/jdg/1349292671.

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5

Tsai, Chung-Jun, Li-Sheng Tseng, and Shing-Tung Yau. "Cohomology and Hodge theory on symplectic manifolds: III." Journal of Differential Geometry 103, no. 1 (May 2016): 83–143. http://dx.doi.org/10.4310/jdg/1460463564.

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6

Crainic, Marius, Rui Loja Fernandes, and David Martínez Torres. "Poisson manifolds of compact types (PMCT 1)." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 756 (November 1, 2019): 101–49. http://dx.doi.org/10.1515/crelle-2017-0006.

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AbstractThis is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

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7

Tardini, Nicoletta, and Adriano Tomassini. "On the cohomology of almost-complex and symplectic manifolds and proper surjective maps." International Journal of Mathematics 27, no. 12 (November 2016): 1650103. http://dx.doi.org/10.1142/s0129167x16501032.

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Let [Formula: see text] be an almost-complex manifold. In [Comparing tamed and compatible symplectic cones and cohomological properties of almost-complex manifolds, Comm. Anal. Geom. 17 (2009) 651–683], Li and Zhang introduce [Formula: see text] as the cohomology subgroups of the [Formula: see text]th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds, we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in [Cohomology and Hodge Theory on Symplectic manifolds: I, J. Differ. Geom. 91(3) (2012) 383–416] by Tseng and Yau and a new characterization of the hard Lefschetz condition in dimension [Formula: see text] is provided.

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8

Yan, Dong. "Hodge Structure on Symplectic Manifolds." Advances in Mathematics 120, no. 1 (June 1996): 143–54. http://dx.doi.org/10.1006/aima.1996.0034.

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9

Lin, Yi. "Hodge theory on transversely symplectic foliations." Quarterly Journal of Mathematics 69, no. 2 (December 14, 2017): 585–609. http://dx.doi.org/10.1093/qmath/hax051.

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10

Mazzeo, Rafe, and Ralph S. Phillips. "Hodge theory on hyperbolic manifolds." Duke Mathematical Journal 60, no. 2 (April 1990): 509–59. http://dx.doi.org/10.1215/s0012-7094-90-06021-1.

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11

Cavalcanti, Gil R. "Hodge theory of SKT manifolds." Advances in Mathematics 374 (November 2020): 107270. http://dx.doi.org/10.1016/j.aim.2020.107270.

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12

Ran, Ziv. "A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION." Journal of the Institute of Mathematics of Jussieu 19, no. 5 (November 9, 2018): 1509–19. http://dx.doi.org/10.1017/s1474748018000464.

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We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

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13

Verbitsky, Misha. "Hodge theory on nearly Kähler manifolds." Geometry & Topology 15, no. 4 (October 28, 2011): 2111–33. http://dx.doi.org/10.2140/gt.2011.15.2111.

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14

Kotschick, D., and S. Schreieder. "The Hodge ring of Kähler manifolds." Compositio Mathematica 149, no. 4 (February 28, 2013): 637–57. http://dx.doi.org/10.1112/s0010437x12000759.

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AbstractWe determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch’s.

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15

Albuquerque, R., and J. Rawnsley. "Twistor theory of symplectic manifolds." Journal of Geometry and Physics 56, no. 2 (February 2006): 214–46. http://dx.doi.org/10.1016/j.geomphys.2005.01.007.

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16

Fu, Lie, and Grégoire Menet. "On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four." Mathematische Zeitschrift 299, no. 1-2 (January 12, 2021): 203–31. http://dx.doi.org/10.1007/s00209-020-02682-7.

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AbstractWe extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

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17

Iwaniec, T., C. Scott, and B. Stroffolini. "Nonlinear Hodge theory on manifolds with boundary." Annali di Matematica Pura ed Applicata 177, no. 1 (December 1999): 37–115. http://dx.doi.org/10.1007/bf02505905.

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18

Hunsicker, E. "Extended Hodge theory for fibred cusp manifolds." Journal of Topology and Analysis 10, no. 03 (August 30, 2018): 531–62. http://dx.doi.org/10.1142/s1793525318500188.

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For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted [Formula: see text] harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted [Formula: see text] harmonic forms are harmonic forms that are almost in the given weighted [Formula: see text] space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted [Formula: see text] case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.

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19

Lupton, Gregory, and John Oprea. "Symplectic manifolds and formality." Journal of Pure and Applied Algebra 91, no. 1-3 (January 1994): 193–207. http://dx.doi.org/10.1016/0022-4049(94)90142-2.

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20

Katzarkov, L., T. Pantev, and B. Toën. "Schematic homotopy types and non-abelian Hodge theory." Compositio Mathematica 144, no. 3 (May 2008): 582–632. http://dx.doi.org/10.1112/s0010437x07003351.

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AbstractWe use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$ on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

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21

Mihai, Ion Alexandru. "Odd symplectic flag manifolds." Transformation Groups 12, no. 3 (August 22, 2007): 573–99. http://dx.doi.org/10.1007/s00031-006-0053-0.

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22

Farajzadeh Tehrani, Mohammad. "Open Gromov–Witten theory on symplectic manifolds and symplectic cutting." Advances in Mathematics 232, no. 1 (January 2013): 238–70. http://dx.doi.org/10.1016/j.aim.2012.09.015.

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23

Ahmed, Zulfikar M., and Daniel W. Stroock. "A Hodge theory for some non-compact manifolds." Journal of Differential Geometry 54, no. 1 (2000): 177–225. http://dx.doi.org/10.4310/jdg/1214342150.

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24

Verbitsky, Misha. "Hyperkähler manifolds with torsion, supersymmetry and Hodge theory." Asian Journal of Mathematics 6, no. 4 (2002): 679–712. http://dx.doi.org/10.4310/ajm.2002.v6.n4.a5.

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25

Entov, Michael, and Leonid Polterovich. "Rigid subsets of symplectic manifolds." Compositio Mathematica 145, no. 03 (May 2009): 773–826. http://dx.doi.org/10.1112/s0010437x0900400x.

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AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

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26

Babenko, I. K., and I. A. Taimanov. "Massey products in symplectic manifolds." Sbornik: Mathematics 191, no. 8 (August 31, 2000): 1107–46. http://dx.doi.org/10.1070/sm2000v191n08abeh000497.

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27

Eliashberg, Yakov. "Book Review: Function theory on symplectic manifolds." Bulletin of the American Mathematical Society 54, no. 1 (August 17, 2016): 135–40. http://dx.doi.org/10.1090/bull/1547.

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28

Boissière, Samuel, Marc Nieper-Wißkirchen, and Alessandra Sarti. "Smith theory and irreducible holomorphic symplectic manifolds." Journal of Topology 6, no. 2 (February 13, 2013): 361–90. http://dx.doi.org/10.1112/jtopol/jtt002.

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29

Urakawa, Hajime. "Yang–Mills Theory over Compact Symplectic Manifolds." Annals of Global Analysis and Geometry 25, no. 4 (June 2004): 365–402. http://dx.doi.org/10.1023/b:agag.0000023246.97746.af.

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30

OKASSA, EUGÈNE. "ON LIE–RINEHART–JACOBI ALGEBRAS." Journal of Algebra and Its Applications 07, no. 06 (December 2008): 749–72. http://dx.doi.org/10.1142/s0219498808003107.

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We show that Jacobi algebras (Poisson algebras respectively) can be defined only as Lie–Rinehart–Jacobi algebras (as Lie–Rinehart–Poisson algebras respectively). Also we show that contact manifolds, locally conformal symplectic manifolds (symplectic manifolds respectively) can be defined only as symplectic Lie–Rinehart–Jacobi algebras (only as symplectic Lie–Rinehart–Poisson algebras respectively). We define symplectic Lie algebroids.

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31

Gritsenko, V., K. Hulek, and G. K. Sankaran. "Moduli spaces of irreducible symplectic manifolds." Compositio Mathematica 146, no. 2 (January 26, 2010): 404–34. http://dx.doi.org/10.1112/s0010437x0900445x.

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AbstractWe study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2]manifolds with polarisation of degree 2dand split type is of general type ifd≥12.

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32

Oprea, John. "Homotopy theory and circle actions on symplectic manifolds." Banach Center Publications 45, no. 1 (1998): 63–86. http://dx.doi.org/10.4064/-45-1-63-86.

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33

Rosensteel, George. "Symplectic manifolds, coadjoint orbits, and mean field theory." International Journal of Theoretical Physics 25, no. 5 (May 1986): 553–59. http://dx.doi.org/10.1007/bf00668789.

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34

Dufour, Jean-Paul, and Aïssa Wade. "On the local structure of Dirac manifolds." Compositio Mathematica 144, no. 3 (May 2008): 774–86. http://dx.doi.org/10.1112/s0010437x07003272.

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AbstractWe give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through m. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

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35

Camere, Chiara. "Lattice polarized irreducible holomorphic symplectic manifolds." Annales de l’institut Fourier 66, no. 2 (2016): 687–709. http://dx.doi.org/10.5802/aif.3022.

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36

SMALE, NAT, and STEVE SMALE. "ABSTRACT AND CLASSICAL HODGE–DE RHAM THEORY." Analysis and Applications 10, no. 01 (January 2012): 91–111. http://dx.doi.org/10.1142/s0219530512500054.

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In previous work, with Bartholdi and Schick [1], the authors developed a Hodge–de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian.

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37

Huybrechts, Daniel. "Birational symplectic manifolds and their deformations." Journal of Differential Geometry 45, no. 3 (1997): 488–513. http://dx.doi.org/10.4310/jdg/1214459840.

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38

Al-Zamil, Qusay S. A., and James Montaldi. "Witten–Hodge theory for manifolds with boundary and equivariant cohomology." Differential Geometry and its Applications 30, no. 2 (April 2012): 179–94. http://dx.doi.org/10.1016/j.difgeo.2011.11.002.

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39

Fukaya, Kenji, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. "Lagrangian floer Theory over integers: spherically positive symplectic manifolds." Pure and Applied Mathematics Quarterly 9, no. 2 (2013): 189–289. http://dx.doi.org/10.4310/pamq.2013.v9.n2.a1.

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40

Tanaka, Hiro Lee, and Li-Sheng Tseng. "Odd sphere bundles, symplectic manifolds, and their intersection theory." Cambridge Journal of Mathematics 6, no. 3 (2018): 213–66. http://dx.doi.org/10.4310/cjm.2018.v6.n3.a1.

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41

Dancer, Andrew, Frances Kirwan, and Andrew Swann. "Implosion for hyperkähler manifolds." Compositio Mathematica 149, no. 9 (June 28, 2013): 1592–630. http://dx.doi.org/10.1112/s0010437x13007203.

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AbstractWe introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of$\mathrm{SU} (n)$actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

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42

Bei, Francesco. "Symplectic manifolds, L-cohomology and q-parabolicity." Differential Geometry and its Applications 64 (June 2019): 136–57. http://dx.doi.org/10.1016/j.difgeo.2019.02.007.

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43

Pym, Brent. "Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets." Compositio Mathematica 153, no. 4 (March 13, 2017): 717–44. http://dx.doi.org/10.1112/s0010437x16008174.

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A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities$\widetilde{E}_{6},\widetilde{E}_{7}$and$\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type$q_{5,1}$are the only log symplectic structures on projective four-space whose singular points are all elliptic.

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44

Karigiannis, Spiro, and Naichung Conan Leung. "Hodge theory for G2 -manifolds: intermediate Jacobians and Abel-Jacobi maps." Proceedings of the London Mathematical Society 99, no. 2 (February 24, 2009): 297–325. http://dx.doi.org/10.1112/plms/pdp004.

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45

DUISTERMAAT, J. J., and A. PELAYO. "COMPLEX STRUCTURES ON FOUR-MANIFOLDS WITH SYMPLECTIC TWO-TORUS ACTIONS." International Journal of Mathematics 22, no. 03 (March 2011): 449–63. http://dx.doi.org/10.1142/s0129167x11006854.

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We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a four-manifold with a symplectic two-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira's class of complex surfaces which admit a nowhere vanishing holomorphic (2,0)-form, but are not a torus nor a K3 surface.

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46

Sozen, Yasar. "A note on Reidemeister torsion and pleated surfaces." Journal of Knot Theory and Its Ramifications 23, no. 03 (March 2014): 1450015. http://dx.doi.org/10.1142/s0218216514500151.

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This paper uses the notion of ℂ-symplectic chain complex and proves an explicit formula for the Reidemeister torsion of an arbitrary ℂ-symplectic chain complex in terms of intersection forms of the homologies. In applications, the formula is applied to closed manifolds and also compact manifolds with boundary by using the homologies with coefficients in complex numbers field. Moreover, an explicit formula for the Reidemeister torsion of representations from the fundamental group of a closed oriented hyperbolic surface to PSL2(ℂ) is presented in terms of the cup product of twisted cohomologies, which is related with Weil–Petersson form and thus the Thurston symplectic form. The formula is also applied to pleated surfaces.

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47

CHIANG, RIVER, and LIAT KESSLER. "CYCLIC ACTIONS ON RATIONAL RULED SYMPLECTIC FOUR-MANIFOLDS." Transformation Groups 24, no. 4 (February 12, 2019): 987–1000. http://dx.doi.org/10.1007/s00031-019-09512-z.

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48

Buse, O., and R. Hind. "Ellipsoid embeddings and symplectic packing stability." Compositio Mathematica 149, no. 5 (March 4, 2013): 889–902. http://dx.doi.org/10.1112/s0010437x12000826.

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AbstractWe prove packing stability for rational symplectic manifolds. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some symplectic volume filling embeddings in dimension 4.

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49

Ginzburg, Viktor L., and Başak Z. Gürel. "Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds." Compositio Mathematica 152, no. 9 (June 17, 2016): 1777–99. http://dx.doi.org/10.1112/s0010437x16007508.

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We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

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50

Amar, Eric. "On the $$L^{r}$$ Hodge theory in complete non compact Riemannian manifolds." Mathematische Zeitschrift 287, no. 3-4 (February 16, 2017): 751–95. http://dx.doi.org/10.1007/s00209-017-1844-9.

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