Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

By Tian-Jun Li, Weiyi Zhang. 2007

124 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/df89c0487dd3847a193916b48777e2658a4932df

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In Geometry and Topology

Citations in the corpus

On cohomological decomposition of almost-complex manifolds and deformations (2009, background): “Zhang proved that if Jis C∞-full and if Kc J is non-empty, then Kt J = K c J + H (2,0),(0,2) J (M)R , where H(2,0),(0,2) J (M)R generalizes the group H2,0 ∂ (M) ⊕H0, 2 ∂ (M) ∩ HdR(M;R) in (1). In [24] dual notions starting from the space of currents are also deﬁned: we will recall in Section 2 wha”
Bott–Chern cohomology of solvmanifolds (2012, background): “(With the notation of [49, 9], this means that, in any case, Γ\G is complex-C∞-pure-and-full at every stage, namely, the de Rham cohomology admits a decomposition in pure-type subgroups with respect to the complex structure.”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “If this result were true, it would have important applications to symplectic topology [21, 42, 43, 54].”
Special Hermitian metrics on compact solvmanifolds (2015, background): “ve results in literature which suggest that Hermitiansymplectic structures on non-Ka¨hler manifolds couldn’t exist. The ﬁrst of these results is about the four dimensional case. Theorem 5.3 (Li-Zhang [37], Streets-Tian [44]). If a compact complex surface admits an Hermitiansymplectic form, then it i”
Ricci Curvatures on Hermitian manifolds (2014, methodology): “, R(1) is the (1,1)-component of the curvature 2-form of the Levi-Civita connection on K−1 M . Hence, if R(1) is strictly positive, it can induce a symplectic structure on M(see Theorem 3.21 and also [36]). Moreover, the symplectic structures thus obtained are not necessarily Ka¨hler. As application”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, background): “Moreover, the study of tamed symplectic structures in dimension 4 is related to a more general conjecture of Donaldson (see for instance [7, 28, 44]).”