Canonical Symplectic Structures and Deformations of Algebraic Surfaces (original) (raw)

Symplectic structures of algebraic surfaces and deformation

2002

Friedman and Morgan made the "speculation" that deformation equivalence and diffeomorphism should coincide for algebraic surfaces. Counterexamples, for the hitherto open case of surfaces of general type, have been given in the last years by Manetti, by Kharlamov-Kulikov and in my cited article. For the latter much simpler examples, it was shown that there are surfaces SSS which are not

Symplectic singularities of varieties: The method of algebraic restrictions

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We study germs of singular varieties in a symplectic space. In [A1], V. Arnol'd discovered so called ''ghost'' symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of di¤erential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of Darboux-Givental' theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We prove that a quasi-homogeneous variety N is contained in a nonsingular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to N vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S 5 singularity, and for regular union singularities.

Local symplectic algebra and simple symplectic singularities of curves

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold in [A1]. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a quasihomogeneous curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semigroups (3, 4, 5), (3, 5, 7), (3, 7, 8). This classification implies Kolgushkin's classification of stably simple symplectic singularities of parameterized curves in the smooth and R-analytic category.

Existence of symplectic surfaces

Geometry and Topology of Manifolds, 2005

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4.

Deformation and duality from the symplectic point of view

2012

In this dissertation we study applications of methods of symplectic topology, mainly the theories of Floer and quantum homology, to the following two subjects in complex projective geometry: (1) Morse-Bott degenerations of Kähler Manifolds and (2) manifolds with small dual. (1) Morse-Bott degenerations: A Morse-Bott degeneration is a holomorphic map π : X → D from a Kähler manifold (X, Ω) to the unit disc D ⊂ C such that Crit(π) ⊂ π−1(0) and the holomorphic Hessian of π is a a non-degenerate quadratic form when restricted to the normal bundle of Crit(π) in X. Our first result on the subject is showing that Lagrangian submanifolds in the singularity locus of the degeneration L ⊂ Crit(π) ⊂ Σ0 give rise to Lagrangian submanifolds Nz(L) in Σz = π −1(z) in the smooth fibers z 6= 0. Moreover, Nz(L) ⊂ Σz is topologically a sphere bundle over L. This could be seen as a generalization of the well known result stating that fibers of a Lefschetz fibration admit Lagrangian spheres corresponding...

Symplectic log Calabi–Yau surface: deformation class

Advances in Theoretical and Mathematical Physics, 2016

We study the symplectic analogue of log Calabi-Yau surfaces and show that the symplectic deformation classes of these surfaces are completely determined by the homological information.

The Smoothability of Normal Crossings Symplectic Varieties

arXiv (Cornell University), 2014

Our previous paper introduces topological notions of normal crossings symplectic divisor and variety and establishes that they are equivalent, in a suitable sense, to the desired geometric notions. Friedman's d-semistability condition is well-known to be an obstruction to the smoothability of a normal crossings variety in a one-parameter family with a smooth total space in the algebraic geometry category. We show that the direct analogue of this condition is the only obstruction to such smoothability in the symplectic topology category. Every smooth fiber of the families of smoothings we describe provides a multifold analogue of the now classical (two-fold) symplectic sum construction; we thus establish an old suggestion of Gromov in a strong form.

Fibred algebraic surfaces and commutators in the Symplectic group

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).

Singularities and semistable degenerations for symplectic topology

Comptes Rendus Mathematique, 2018

We overview our work [6, 7, 8, 9, 10] defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.