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Register a new userSymplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
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For $(mathbb{C} P^2 # 5{overline {mathbb{C} P^2}},omega)$, let $N_{omega}$ be the number of $(-2)$-symplectic spherical homology classes.We completely determine the Torelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if $N_{omega}>8$; it is $pi_0(Diff^+(S^2,5))$ if $N_{omega}=0$ (by Paul Seidel and Jonathan Evans); it is $pi_0(Diff^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $pi_1(Symp(mathbb{C} P^2 # 5{overline {mathbb{C} P^2}}, omega)$ for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type $mathbb{A}$ and type $mathbb{D}$ Lie algebras. We also provide a solution to the smooth isotopy problem of rational $4$-manifolds.
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We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohomologies on symplectic manifolds. Using these operators, we introduce new primitive cohomologies that are analogous to the Dolbeault cohomology in the complex theory. Interestingly, the finiteness of these primitive cohomologies follows directly from an elliptic complex. We calculate the known primitive cohomologies on a nilmanifold and show that their dimensions can vary depending on the class of the symplectic form.
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $mathbb{C}P^2# Noverline{mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with non-negative self-intersection, following the strategy of McDuffs classification of rational and ruled symplectic four manifolds.
We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of the primitive cohomologies of differential forms. We provide a general algorithm for computing the monodromies of the fibrations explicitly, which are needed to determine the primitive cohomologies. We also investigate a similar phenomenon coming from fibrations of a class of graph links, whose primitive cohomology provides information about the fibration structure.
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