Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds

In this article we construct an infinite family of simply connected minimal symplectic 4-manifolds, each of which admits at least two nonisomorphic Lefschetz fibration structures with the same generic fiber. We obtain such examples by performing knot surgery on an elliptic surface E(n) using a special type of 2-bridge knots. This work was supported by grant No. R01-2005-000-10625-0 from the KOSEF and by the Korea Research Foundation Grant funded by the Korean Government (KRF-2007-314-C00024).     

Poisson cohomology of broken Lefschetz fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.     

Cyclic cohomology and the family Lefschetz theorem

Any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra. We prove that such cocycle furnishes additive maps from the vertically equivariant K-theory to the scalars. This enables to associate to any closed current on the base of the fibration, a Lefschetz formula for fiber-preserving isometries. Using geometric operators on the base, we deduce the integrality of some characteristic numbers. Received: 28 June 2001 / Published online: 1 February 2002     

A weak version of the Lipman–Zariski conjecture

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.     

Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map

We construct a spectral sequence converging to symplectic homology of a Lefschetz fibration whose E ¹ page is related to Floer homology of the monodromy symplectomorphism and its iterates. We use this to show the existence of fixed points of certain symplectomorphisms.     

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

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 = c₁ ²/c₂ do not admit gaps — following similar results of Sommese in the complex case.     

Homological mirror symmetry for Brieskorn�CPham singularities

We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn–Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an abelian group of rank one. Symplectic Picard-Lefschetz theory developed by Seidel is an essential ingredient of the proof.     

Chern-Connes character for the invariant Dirac operator in odd dimensions

In this paper we give a proof of the Lefschetz fixed point formula of Freed for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced by Lafferty et al. and then we generalize this formula under the noncommutative geometry framework.     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E₂. In the appendix, we prove nontriviality of certain Gromov-Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of  along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.     

Chern numbers of certain Lefschetz fibrations

We address the geography problem of relatively minimal Lefschetz fibrations over surfaces of nonzero genus and prove that if the fiber-genus of the fibration is positive, then (equivalently, ) holds for those symplectic 4-manifolds. A useful characterization of minimality of such symplectic 4-manifolds is also proved.

     

Meyer's signature cocycle and hyperelliptic fibrations

We show that the cohomology class represented by Meyer's signature cocycle is of order in the 2-dimensional cohomology group of the hyperelliptic mapping class group of genus . By using the -cochain cobounding the signature cocycle, we extend the local signature for singular fibers of genus 2 fibrations due to Y. Matsumoto [18] to that for singular fibers of hyperelliptic fibrations of arbitrary genus and calculate its values on Lefschetz singular fibers. Finally, we compare our local signature with another local signature which arises from algebraic geometry. Received: 6 August 1998 / in final form: 24 February 1999     

Hyperplane sections and the subtlety of the Lefschetz properties

The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all characteristics, whereas a special hyperplane section does not have this property in any characteristic.     

Regularity of a dynamic neighborhood of a regular language

Birational maps give the main research tool for the theory of Fano varieties, as we know from the fundamental works of V.A. Iskovskikh and his school. Nowadays one can exploit them in the new approach of D. Auroux to Mirror Symmetry of Fano varieties, which is based on a certain generalization of the notion of special Lagrangian fibration suitable for Fano varieties. We present a very simple example of how a special Lagrangian fibration can be transferred by a birational map. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 209–211.     

Surface Bundles with Nonvanishing Signature

Using the theory of Lefschetz fibrations and recent advances in mapping class group theory, surface bundles over surfaces with nonzero signature and small base genus are constructed. In particular, a genus-5 fibration over the surface of genus 26 with nonzero signature is given –- improving former results on the possible base genera for surface bundles over surfaces with nonzero signature.     

LANGLANDS DUALITY AND G2 SPECTRAL CURVES

We first demonstrate how duality for the fibres of the so-calledHitchin fibration works for the Langlands dual groups Sp(2m)and SO(2m + 1). We then show that duality for G₂ is implementedby an involution on the base space which takes one fibre toits dual. A formula for the natural cubic form is given andshown to be invariant under the involution.     

On Lefschetz periodic point free self-maps

We study the periodic point free maps and Lefschetz periodic point free maps on connected retract of a finite simplicial complex using the Lefschetz numbers. We put special emphasis in the self-maps on the product of spheres and of the wedge sums of spheres.     

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.