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    Fujiki relations and fibrations of irreducible symplectic varieties
    Épijournal de Géométrie Algébrique (2020)
    Martin Schwald
    This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds.
    Rank (graph theory)
    Citations (11)
    Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds
    arXiv (Cornell University) (2007)
    Tianjun LiWeiyi Zhang
    We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and complex surfaces.
    Duality (order theory)
    Homology
    Citations (4)
    On symplectically aspherical manifolds with nontrivial pi_2
    arXiv (Cornell University) (1998)
    Robert E. Gompf
    We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the conditions that these two cohomology classes vanish on spherical homology. In particular, we show that the symplectic form can pair trivially with all spherical classes even when the Chern class pairs nontrivially.
    Homology
    Characteristic class
    Symplectic vector space
    Citations (0)
    Hermitian Metrics on Compact Complex Manifolds and Their Deformation Limits
    Springer INdAM series (2017)
    Antonio OtalLuis UgarteRaquel Villacampa
    Hermitian symmetric space
    Citations (5)
    Lefschetz pencils and the canonical class for symplectic four-manifolds
    Topology (2003)
    Simon DonaldsonIvan Smith
    Symplectomorphism
    Argument (complex analysis)
    Manifold (fluid mechanics)
    Citations (62)
    Symplectic harmonicity and generalized coeffective cohomologies
    Luis UgarteRaquel Villacampa
    Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by C.-J. Tsai, L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms $\omega_t$ such that the dimensions of these symplectic cohomology groups vary with respect to $t$. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an $8$-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.
    Harmonic
    Citations (0)
    Tamed symplectic forms on complex manifolds
    arXiv (Cornell University) (2007)
    Tianjun LiWeiyi Zhang
    We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and complex surfaces.
    Duality (order theory)
    Homology
    Citations (4)
    Lefschetz pencils and the canonical class for symplectic 4-manifolds
    arXiv (Cornell University) (2000)
    Simon DonaldsonIvan Smith
    We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) > 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X may be represented by an embedded symplectic submanifold. The result builds on the existence of Lefschetz pencils on symplectic four-manifolds. We approach the topological problem of constructing submanifolds with locally positive intersections via almost complex geometry. The crux of the argument is that a Gromov invariant counting pseudoholomorphic sections of an associated bundle of symmetric products is non-zero.
    Symplectomorphism
    Citations (16)
    On symplectic quotients of K3 surfaces
    Indagationes Mathematicae (2000)
    Zübeyir ÇınkırHurşit Önsiper
    K3 surface
    Symplectic group
    Citations (4)