Symplectic four-manifolds and Lefschetz fibrations

After blowing up at isolated points, a symplectic four-manifold admits a Lefschetz fibration over the sphere, thus expressing the manifold as a family of Riemann sufaces (finitely many of which have nodal singularities).  If almost complex structures are chosen appropriately pseudoholomorphic curves in the manifold can be identified with certain sections of a bundle of symmetric products associated to the Lefschetz fibration.  Simon Donaldson and Ivan Smith defined a "standard surface count" which enumerates such sections, and while they did not directly connect this invariant to pseudoholomorphic curves in the original manifold they used its properties to reprove a result of Taubes concerning the existence of symplectic surfaces Poincare dual to the canonical class  (a result that Taubes had proven using Seiberg-Witten theory).  This paper proves that, under certain satisfiable assumptions on the Lefschetz fibration, the Donaldson-Smith invariant is equal to the Gromov-Taubes invariant enumerating pseudoholomorphic curves in the original manifold (which Taubes had in turn earlier shown to equal the Seiberg-Witten invariant).  This implies that the Donaldson-Smith invariant does not depend on the choice of Lefschetz fibration structure, and, thanks to an earlier result of Smith based on a parametrized version of Serre duality, leads to a Seiberg-Witten-free proof of the "Taubes duality" relation between the Gromov invariants associated to different homology classes.  

This paper represents an attempt to extend the Donaldson-Smith approach to more general types of pseudoholomorphic curves.  The Gromov-Taubes invariant is an integer-valued count of embedded, possibly disconnected pseudoholomorphic curves (possibly with multiply-covered square-zero torus components); if one attempts to count singular curves in a similar way one potentially encounters problems arising from excess-dimensional boundary components.  Here we transfer the problem to the Donaldson-Smith picture, and use a family blowup approach to obtain an integer-valued count FDS of sections of a symmetric product fibration which, at least heuristically, correspond to nodal curves in the original four-manifold.  In special cases where a "nodal Gromov-Taubes invariant" can be straightforwardly defined, FDS coincides with it; however FDS can be defined in a more general setting and, conjecturally, is an invariant of the manifold and not just of the Lefschetz fibration.

 This is a survey article describing the results of the two papers listed above.

This paper fits the Donaldson-Smith invariant into a (restricted) "field theory" setup, wherein one associates to any 3-manifold fibered over the circle (together with some additional data) a Floer homology group, and to any Lefschetz fibration over the annulus a map between the groups associated to the boundary components; the Donaldson-Smith invariant of the Lefschetz fibration is then recovered as the degree of the map associated to the restriction of the  fibration over the complement of two small discs.  The relevant Floer homology groups are adapted from a construction of Dietmar Salamon using the symplectic vortex equations.  These equations involve a parameter tau, and an asymptotic analysis as tau tends to infinity shows that if the group associated to a fibered three-manifold does not vanish then the monodromy map must have periodic points of a prescribed period.  Many parallels are exhibited between this field theory and the ones arising from Seiberg-Witten Floer and Heegaard Floer theories; presumably appropriate versions of these are isomorphic.

While there is still no proof in the literature that the Floer groups discussed here are isomorphic to the Seiberg-Witten Floer groups, closely related statements are now known, as Yi-Jen Lee and Cliff Taubes have shown that Periodic Floer Homology is isomorphic to Seiberg-Witten Floer homology.  Also, Tim Perutz developed a similar field theory independently around the same time, and his theory extends to "broken Lefschetz fibrations," which exist on all oriented four-manifolds and not just symplectic ones; moreover, Perutz and Yanki Lekili have work in progress connecting this theory to Heegaard Floer theory.