Related papers: Fukaya A_\infty-structures associated to Lefschetz…

We give a maximal set of disjoint $(-1)$-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz…

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In…

The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are…

A class of partially wrapped Fukaya categories in $T^* N$ are proven to be well defined and then studied. In the case of $N$ diffeomorphic to $\mathbb{R}^m \times \mathbb{T}^n$, it is shown that these categories provide homological mirrors…

We give an introduction to partially wrapped Fukaya categories of surfaces with orbifold singularities. Dissecting an orbifold surface $\mathbf S$ into polygons, certain dissections give rise to formal generators, inducing a triangulated…

In this paper we discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumption, the (smooth)…

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive…

We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that the (pluri)canonical pencils on these surfaces have the same monodromy groups, and are related by a…

In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk…

For a symplectic manifold satisfying some topological condition,we define a special class of modules over the deformation quantization algebra. For any two such modules we construct an infinity local system of morphisms. We construct such…

To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is the complement…

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the…

We prove that any genus-2 Lefschetz fibration without reducible fibers and with ``transitive monodromy'' is holomorphic. The latter condition comprises all cases where the number of singular fibers is not congruent to 0 modulo 40. An…

Let M be a Weinstein four-manifold mirror to Y\D for (Y,D) a log Calabi--Yau surface; intuitively, this is typically the Milnor fibre of a smoothing of a cusp singularity. We introduce two families of symplectomorphisms of M: Lagrangian…

As an explicit example of an $A_\infty$-structure associated to geometry, we construct an $A_\infty$-structure for a Fukaya category of finitely many lines (Lagrangians) in $\R^2$, ie., we define also {\em non-transversal}…

We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove…

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has…

In this paper we establish a version of homological mirror symmetry for punctured Riemann surfaces. Following a proposal of Kontsevich we model A-branes on a punctured surface $\Sigma$ via the topological Fukaya category. We prove that the…

In this paper we construct and classify Lagrangian T^3-fibrations on non compact symplectic manifolds with singular fibres of prescribed topological type. This contributes to the understanding of the structure of the singular fibres that…

We prove that, for any Morse function on a compact manifold and any adapted gradient satisfying the Morse-Smale condition, there is a homotopically unique complex-valued symplectic Lefschetz fibration on the cotangent bundle whose…