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We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward and Meinrenken, and show it can be interpreted in algebro-geometric terms. A key ingredient is the `universal cut of the cotangent bundle of the group itself, which is identified with a moduli space of framed bundles on chains of projective lines recently introduced by the authors.
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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 study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K rtimes S_2$ where $K < SL_2(C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $dim V eq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.
Given a symplectic three-fold $(M,omega)$ we show that for a generic almost complex structure $J$ which is compatible with $omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $ggeq 0$ representing a homology class $beta$ in $H_2(M,Z)$ with $c_1(M).beta=0$, provided that the divisibility of $beta$ is at most 4 (i.e. if $beta=nalpha$ with $alphain H_2(M,Z)$ and $nin Z$ then $nleq 4$). Moreover, each such curve is embedded and 4-rigid.
We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety $X$, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification $(M,mathbf{D})$ of $X$. We exhibit a broad class of pairs $(M,mathbf{D})$ (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective $M$, of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work [GP1].
In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This exten
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