Symplectic Lefschetz Fibrations Research Articles

Geography of symplectic Lefschetz fibrations and rational blowdowns

We produce simply-connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic 4 4 –manifolds. We present an example of a construction of a minimal symplectic exotic C P 2 # 5 C P ¯ 2 {\mathbb {CP}}{}^{2}\# \, 5 \overline {\mathbb {CP}}{}^{2} through this procedure applied to a genus– 3 3 fibration.

Lefschetz fibrations with arbitrary signature

We develop techniques to construct explicit symplectic Lefschetz fibrations over the $2$-sphere with any prescribed signature $\sigma$ and any spin type when $\sigma$ is divisible by $16$. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to connected sums of $S^2 \times S^2$, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.

Compactifications of adjoint orbits and their Hodge diamonds

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.

Some Landau–Ginzburg models viewed as rational maps

Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.

Symplectic Lefschetz fibrations on adjoint orbits

Abstract We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We describe the topology of the regular and singular fibres, in particular we calculate their middle Betti numbers.

Symplectic Lefschetz fibrations of curves as gonal coverings

We show that after a finite base change every symplectic Lefschetz fibration \({f \colon X \rightarrow B}\) of genus g > 3 curves over a closed oriented surface becomes a finite covering of degree \({\frac{g}{2} + 1}\) or \({\frac{g}{2} + \frac{3}{2}}\) of a family of spheres over a Riemann surface, with a branch locus admitting complex algebraic curves as local models. In the case of fibers of genus 4, it is shown that after a 2:1 base change the family admits a trigonal covering to a symplectic ruled surface, with symplectic branch locus.

Strongly fillable contact manifolds and J-holomorphic foliations

We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion.

On fundamental groups of symplectically aspherical manifolds II: Abelian groups

We describe all abelian groups which can appear as the fundamental groups of closed symplectically aspherical manifolds. The proofs use the theory of symplectic Lefschetz fibrations.

Singular Lefschetz pencils

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality

We prove upper bounds for the number of critical points in semi- stable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparating vanishing cycles in Lefschetz pencils and reprove the known lower bounds for the commutator lengths of Dehn twists.

Hyperelliptic Szpiro Inequality

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus g, the number N of nonseparating vanishing cycles and the number D of singular fibers satisfy the inequality N ≤ (4g + 2)D.

On the topology of compact Stein surfaces

The main result of this paper is that any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over S2. Also along the way we obtain the following results. (1) There exists a minimal elliptic fibration over D2, which is not Stein. (2) The circle bundle over a genus n ≥ 2 surface with Euler number e = −1 admits at least n + 1 mutually nonhomeomorphic simply-connected Stein fillings. (3) Any surface bundle over S1, whose fiber is a closed surface of genus n ≥ 1 can be embedded into a closed symplectic 4-manifold, splittingthe symplectic 4-manifoldinto two pieces both of which have positive b2+⁠. (4) Every closed, oriented, connected 3-manifold has a weakly symplectically fillable double cover, branched along a 2-component link.

Symplectic Lefschetz fibrations onS1×M3

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle.

Indecomposability of certain Lefschetz fibrations

We prove that Lefschetz fibrations admitting a section of square -1 cannot be decomposed as fiber sums. In particular, Lefschetz fibrations on symplectic 4-manifolds found by Donaldson are indecomposable. This observation also shows that symplectic Lefschetz fibrations are not necessarily fiber sums of holomorphic ones.

Symplectic Lefschetz fibrations with arbitrary fundamental groups

In this paper we give an explicit construction of a symplectic Lefschetz fibration whose total space is a smooth compact four dimensional manifold with a prescribed fundamental group. We also study the numerical properties of the sections in symplectic Lefschetz fibrations and their relation to the structure of the monodromy group.

Harmonic Maps to Teichmüller Space

We give sufficient conditions for the existence of equivariant harmonic maps from the universal cover of a Riemann surface B to the Teichmuller space of a genus g ≥ 2 surface Σ. The condition is in terms of the representation of the fundamental group of B to the mapping class group of Σ. The metric on Teichmuller space is chosen to be the Kahler hyperbolic metric. Examples of such representations arise from symplectic Lefschetz fibrations. Date: January 30, 2000 (final revision). 1991 Mathematics Subject Classification. Primary: 58E20 ; Secondary: 32G13, 30F60, 57M60. G.D. supported in part by NSF grant DMS-9803606. L.K. supported in part by NSF grant DMS-9700605 and a Sloan Research Fellowship. R.W. supported in part by NSF grant DMS-9971860.

The Geography Problem of 4-Manifolds with Various Structures

We review results about the geography problem of complex, symplectic and Einstein 4-manifolds. Finally we discuss the same problem for Lefschetz fibrations. We show that in the geography of symplectic 4-manifolds and Lefschetz fibrations the slopes α = c12/c2 do not admit gaps — following similar results of Sommese in the complex case.

ON HYPERELLIPTIC C∞–LEFSCHETZ FIBRATIONS OF FOUR-MANIFOLDS

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces.