Mathematics

Appropriate submanifolds of a Poisson manifold have an induced Poisson structure. The first part of this paper discusses how certain Poisson-theoretic properties of such submanifolds and the ambient Poisson manifold are related; a distinguished rôle will be played by the so-called coregular submanifolds. The main objective of this paper is to analyze in detail Poisson submersions whose fibers are coregular submanifolds, and to present instances in which such coregular Poisson submersions appear.

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency constraints. We give a formula describing how these invariants arise as point constraints are pushed together in dimension four, and we use this to recursively compute all of these invariants in terms of Gromov--Witten invariants of blowups. As a key tool, we study analogous invariants which count punctured curves with negative ends on a small skinny ellipsoid.

Based on computations of Pandharipande, Zinger proved that the Gopakumar-Vafa BPS invariants BPS A,g (X,ω) for primitive Calabi-Yau classes and arbitrary Fano classes A on a symplectic 6 -manifold (X,ω) agree with the signed count n A,g (X,ω) of embedded J -holomorphic curves representing A and of genus g for a generic almost complex structure J compatible with ω . Zinger's proof of the invariance of n A,g (X,ω) is indirect, as it relies on Gromov--Witten theory. In this article we give a direct proof of the invariance of n A,g (X,ω) . Furthermore, we prove that n A,g (X,ω)=0 for g≫1 , thus proving the Gopakumar-Vafa finiteness conjecture for primitive Calabi-Yau classes and arbitrary Fano classes.

We prove a relationship between quantum Steenrod operations and the quantum connection. In particular there are operations extending the quantum Steenrod power operations that, when viewed as endomorphisms of equivariant quantum cohomology, are covariantly constant. We demonstrate how this property is used in computations of examples.

Let (M,ω) be a ruled symplectic four-manifold. If (M,ω) is rational, then every homologically trivial symplectic cyclic action on (M,ω) is the restriction of a Hamiltonian circle action.

Let (X,ω) be a compact symplectic manifold whose first Chern class c 1 (X) is divisible by a positive integer n . We construct a Z 2n -action on its Fukaya category Fuk(X) and a Z n -action on the local models of its moduli of Lagrangian branes. We show that this action is compatible with the gluing functions for different local models.

We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding S 1 -equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (non-equivariant) open-closed map is. These {\em cyclic open-closed maps} give (a) constructions of geometric smooth and/or proper Calabi-Yau structures on Fukaya categories (which in the proper case implies the Fukaya category has a cyclic A-infinity model in characteristic 0) and (b) a purely symplectic proof of the non-commutative Hodge-de Rham degeneration conjecture for smooth and proper subcategories of Fukaya categories of compact symplectic manifolds. Further applications of cyclic open-closed maps, to counting curves in mirror symmetry and to comparing topological field theories, are the subject of joint projects with Perutz-Sheridan [GPS1, GPS2] and Cohen [CG].

We redefine the cord algebra, which was introduced by Lenhard Ng as a topological knot invariant, in terms of Morse Theory. The determination of the cord algebra of the unknot and of the righthanded trefoil are given. We proove that the cord algebra in our definition is a knot invariant.

In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence of Kähler metrics of constant scalar curvature. In the third part of the paper we outline recent results of both authors, and in particular show that the constant scalar curvature Kähler metric problem and the study of deformation quantization meet at the notion of trace (density) for star product. We formulate a cohomology formula for the invariant of K-stability condition on Kähler metrics with constant Cahen-Gutt momentum.

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.