Symplectic rational $G$-surfaces and equivariant symplectic cones

Journal of Algebra, 2020

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, constructing a fibration with s = 1 and irreducible singular fibre with 4 nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group Sp(2g, Z).

2002

We show that a surface of general type has a canonical symplectic structure (up to symplectomorphism) which is invariant for smooth deformation. Our main theorem is that the symplectomorphism type is also invariant for deformations which also allow certain normal singularities, called Single Smoothing Singularities ( and abbreviated as SSS), or yielding Q-Gorenstein smoothings of quotient singularities. Using the counterexamples of M.Manetti to the DEF = DIFF question whether deformation type and diffeomorphism type coincide for algebraic surfaces, we show that these yield surfaces of general type which are not deformation equivalent but are symplectomorphic. In particular, they are diffeomorphic through a diffeomorphism carrying the canonical class of one to the canonical class of the other surface. ii) Another interesting corollary is the existence of cuspidal algebraic plane curves which are symplectically isotopic, but not equisingular deformation equivalent. * The research of the author was performed in the realm of the SCHWERPUNKT "Globale Methode in der komplexen Geometrie", and of the EAGER EEC Project.

Communications in Contemporary Mathematics, 2009

We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.

Journal of the American Mathematical Society

Let M M be either S 2 × S 2 S^2\times S^2 or the one point blow-up C P 2 # C P ¯ 2 {\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2 of C P 2 {\mathbb {C}}P^2 . In both cases M M carries a family of symplectic forms ω λ \omega _{\lambda } , where λ > − 1 \lambda > -1 determines the cohomology class [ ω λ ] [\omega _\lambda ] . This paper calculates the rational (co)homology of the group G λ G_\lambda of symplectomorphisms of ( M , ω λ ) (M,\omega _\lambda ) as well as the rational homotopy type of its classifying space B G λ BG_\lambda . It turns out that each group G λ G_\lambda contains a finite collection K k , k = 0 , … , ℓ = ℓ ( λ ) K_k, k = 0,\dots ,\ell = \ell (\lambda ) , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as λ → ∞ \lambda \to \infty . However, for each fixed λ \lambda there is essentially one nonvanishing product that g...

Izvestiya: Mathematics, 2005

We prove that a resolution of singularities of any finite covering of the projective plane branched along a Hurwitz curveH and, maybe, along a line "at infinity" can be embedded as a symplectic submanifold into some projective algebraic manifold equipped with an integer Kähler symplectic form (assuming that ifH has negative nodes, then the covering is non-singular over them). For cyclic coverings we can realize this embeddings into a rational algebraic 3-fold. Properties of the Alexander polynomial ofH are investigated and applied to the calculation of the first Betti number b 1 (X n) of a resolution X n of singularities of n-sheeted cyclic coverings of CP 2 branched alongH and, maybe, along a line "at infinity". We prove that b 1 (X n) is even ifH is an irreducible Hurwitz curve but, in contrast to the algebraic case, that it can take any non-negative value in the case whenH consists of several irreducible components.

arXiv: Symplectic Geometry, 2019

We apply Zhang's almost Kahler Nakai-Moishezon theorem and Li-Zhang's comparison of $J$-symplectic cones to establish a stability result for the symplectomorphism group of a rational surface with Euler number up to $12$.

arXiv: Symplectic Geometry, 2019

We survey the progresses on the study of symplectic geometry past four decades. We briefly deal with the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.

arXiv (Cornell University), 2008

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T 2 -fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kähler surfaces raised in .

Geometry & Topology, 2012

Given a Lagrangian sphere in a symplectic 4-manifold .M; !/ with b C D 1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension Ä of .M; !/ is 1, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case. On the existence side, when Ä D 1, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Bulletin des Sciences Mathématiques, 2009

Let K C be the complexification of a compact connected Lie group K. Fixing a K-invariant inner product on Lie(K), the total space of T * K is identified with K C. We show that the Liouville symplectic form on T * K is Kähler with respect to the complex structure of K C .