Results 1  10
of
94
INTEGRATION OVER THE uPLANE IN DONALDSON THEORY
, 1997
"... We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be a ..."
Abstract

Cited by 64 (3 self)
 Add to MetaCart
We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the uplane integrals, the relation of Donaldson theory to N = 2 supersymmetric YangMills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to fourmanifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.
Product formulas along T 3 for SeibergWitten invariants
 Math. Res. Letters
, 1997
"... Suppose that X is a smooth closed oriented 4manifold, and that X contains a smoothly embedded 2torus T 2 ↩ → X with trivial selfintersection number. Similarly to Dehnsurgery on knots in 3manifolds, a generalized logarithmic ..."
Abstract

Cited by 38 (1 self)
 Add to MetaCart
Suppose that X is a smooth closed oriented 4manifold, and that X contains a smoothly embedded 2torus T 2 ↩ → X with trivial selfintersection number. Similarly to Dehnsurgery on knots in 3manifolds, a generalized logarithmic
Minimal entropy and collapsing with curvature bounded from below
 Invent. Math
"... Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting ..."
Abstract

Cited by 37 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting an Fstructure is zero. We also show that if M admits an Fstructure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is nonnegative. We show that Fstructures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, CP 2, CP 2, S 2 × S 2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4, CP 2, S 2 × S 2, CP 2 #CP 2 or CP 2 #CP 2. Finally, suppose that M is a closed simply connected 5manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5, S 3 ×S 2, the nontrivial S 3bundle over S 2 or the Wumanifold SU(3)/SO(3). 1.
On the moduli space of diffeomorphic algebraic surfaces
 Invent. Math
"... Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic descripti ..."
Abstract

Cited by 35 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic description of rational blow down, classical BrillNoether theory and deformation theory of normal flat abelian covers. One of the main problems concerning the differential topology of algebraic surfaces leaving unsolved by the “SeibergWitten revolution ” was to determine whether the differential type of a compact complex surface determines the deformation type. Two compact complex manifolds have the same deformation type if they are fibres of a proper smooth family over a connected
FAMILY BLOWUP FORMULA, ADMISSIBLE GRAPHS AND THE ENUMERATION OF SINGULAR CURVES, I
"... In this paper, we discuss the scheme of enumerating the singular holomorphic curves in a linear system on an algebraic surface. Our approach is based on the usage of the family SeibergWitten invariant and tools from differential topology and algebraic geometry. In particular, one shows that the num ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
(Show Context)
In this paper, we discuss the scheme of enumerating the singular holomorphic curves in a linear system on an algebraic surface. Our approach is based on the usage of the family SeibergWitten invariant and tools from differential topology and algebraic geometry. In particular, one shows that the number of δnodes nodal curves in a generic δ dimensional sublinear system can be expressed as a universal degree δ polynomial in terms of the four basic numerical invariants of the linear system and the algebraic surface. The result enables us to study in detail the structure of these enumerative invariants. 1.
Symplectic fillings and positive scalar curvature
, 1998
"... Let X be a 4–manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b + 2 (X)> 0 or the bo ..."
Abstract

Cited by 31 (10 self)
 Add to MetaCart
Let X be a 4–manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b + 2 (X)> 0 or the boundary of X is disconnected. As an application we show that the Poincaré homology 3–sphere, oriented as the boundary of the positive E8 plumbing, does not carry symplectically semifillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3–manifold which is not symplectically semifillable. Using work of Frøyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3–spheres having positive scalar curvature metrics.
Symplectic convexity in lowdimensional topology
 Proceedings of the Georgia Topology Conference
, 1998
"... Abstract. In this paper we will survey the the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity inlow dimensional topology. ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we will survey the the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity inlow dimensional topology.
Seiberg–Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers
 Turkish J. of Math
, 1999
"... Let f: Σ → Σ be an orientation preserving diffeomorphism of a compact oriented Riemann surface. This paper relates the SeibergWitten invariants of the mapping torus Yf to the Lefschetz invariants of f. 1 ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
Let f: Σ → Σ be an orientation preserving diffeomorphism of a compact oriented Riemann surface. This paper relates the SeibergWitten invariants of the mapping torus Yf to the Lefschetz invariants of f. 1