arXiv research

Embeds cohomology of hyperkahler manifolds into torus cohomology.

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…

The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integ…

Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to e…

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the …

In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber ge…

Researchers find multiple supporting minimal Lefschetz pencils for a complex fibration.

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to th…

We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.

The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.

The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.

Constructs Lefschetz fibrations with slopes near 2.

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above …

Study 1-Lefschetz contact solvmanifolds, proving their characterization.

Jones polynomials compute weighted sums of Lefschetz numbers.

The harmonic cohomology of a Donaldson symplectic submanifold and of an Auroux symplectic submanifold are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $s$-Lefschetz propery. In particular, we consider the symplectic blow-ups of the comp…

The paper embeds 4-manifolds into CP^2 x CP^1 using Lefschetz fibrations.

The paper extends trisection construction for Lefschetz fibrations with $(-n)$-sections.

Proves existence of regular Lagrangians via Weinstein Lefschetz fibrations.

New proof of indecomposable minimal genus-2 Lefschetz fibration.

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 with certain non-trivial global monodromies using chart descriptions, and identify…

The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.

The paper proves a generalized Lefschetz duality for a specific type of manifold.

A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…

The paper defines simplified trisections and provides a mapping class group criterion for their diagrams.

Study calculates Poisson cohomology of broken Lefschetz fibrations.

In this paper we study the behaviour of the Lefschetz property under the blow-up construction. We show that it is possible to reduce the dimension of the kernel of the Lefschetz map if we blow up along a suitable submanifold satisfying the Lefschetz property. We use that, together with results about Massey products, to…

This paper provides a new proof of the Lefschetz fixed point formula using groupoids.

Simplicial spheres have a weak Lefschetz property in characteristic 2.

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.

We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz …

The study shows that symplectic Lefschetz fibrations can have infinitely many sections.

Proves existence of Lefschetz fibrations with arbitrary slopes.

Spin Lefschetz fibrations can represent any group and lattice point.

Constructs genus-3 Lefschetz fibrations and exotic 4-manifolds with specific $b_2^+$ values.

Authors construct symplectic Lefschetz pencils on complex projective plane.

Reviews g-theorem and hard Lefschetz theorem for face rings.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for genus-two Lefschetz fibrations, and show that any genus-two Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.

Study embedding of achiral Lefschetz fibrations in 6D manifolds.

Paper proves existence of compatible Lefschetz fibrations on 6-ball and Stein domains.

The existence of a positive allowable Lefschetz fibration on a compact Stein surface with boundary was established by Loi and Piergallini by using branched covering techniques. Here we give an alternative simple proof of this fact and construct explicitly the vanishing cycles of the Lefschetz fibration, obtaining a dir…

We classify genus 2 Lefschetz fibrations on certain 4-manifolds.

This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topo…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

Constructs inequivalent Lefschetz fibrations on various surfaces.