Smooth Four-Manifolds and Complex Surfaces

Friedman, Robert; Morgan, John

doi:10.1007/978-3-662-03028-8

Cited by 278 publications

(357 citation statements)

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“…Assuming that T m < 2 we see that again the operator B must be equal to one (see [21], [25] for mathematical proof of this result). The net effect of our manipulations is the replacement of the intersecting cycles on the manifold X by the non-intersecting cycles on the manifoldX.…”

Section: Specific Computations Of the Contact Termsmentioning

confidence: 88%

Testing Seiberg-Witten Solution

Losev

Nekrassov

Shatashvili

1999

Strings, Branes and Dualities

123

166

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Abstract. We propose a few tests of Seiberg-Witten solutions of N = 2 supersymmetric gauge theories by the instanton calculus in twisted gauge theories. We re-examine the low-energy effective abelian theory in the presence of sources and present the formalism which makes duality transformations transparent and easily fixes all the contact terms in a broad class of theories. We also discuss ADHM integration and its relevance to the stated problems.

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Section: Specific Computations Of the Contact Termsmentioning

confidence: 88%

Testing Seiberg-Witten Solution

Losev

Nekrassov

Shatashvili

1999

Strings, Branes and Dualities

123

166

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“…This theorem was first proved for the SU(2) Donaldson invariants of simply connected manifolds by Morgan, Mrowka and Ruberman (unpublished). That case is also covered by the results of [6] and by (4) and the proofs can probably be extended to the general case. However, our argument is much simpler, based, as it is, on a simple observation about the orientations of SO(3) moduli spaces [13].…”

Section: Gv)>\$?-o-(x)\-b 2 (X)mentioning

confidence: 93%

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Kotschick

Matić

1995

Math. Proc. Camb. Phil. Soc.

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IntroductionOne of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula There are a number of results on this question in the literature. They can be divided into two classes, those proved by classical topological methods, which apply to topologically locally flat embeddings, including the G-signature theorem, and those proved by methods of gauge theory, which apply to smooth embeddings only.On the classical side, there is the result of Kervaire and Milnor[10], based on Rokhlin's theorem, which shows that certain homology classes are not represented by spheres. A major step forward was made by Rokhlin[20] and Hsiang and Szczarba [8] who introduced branched covers to study this problem for divisible homology classes. If the integral homology class represented by an embedded surface £ c: X is divisible by k, then the corresponding cover of X of order k branched along £ is again a smooth 4-manifold, so there is an obvious inequality, simply from the existence of the covering, asserting that its second Betti number is at least the absolute value of its signature. In the case k = 2, this gives

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“…The spectral cover. In this section we give a construction of the spectral cover similar to the one described in [12], [14] (sections 4.3 and 5.1) and [4].…”

Section: Elliptic Fibrations and Relative Fourier-mukai Transformmentioning

confidence: 99%

“…Friedman and Morgan ( [11] Th.3.3 or [12]) and O'Grady ( [20] Proposition I.1.6) have proved that for vector bundles of positive relative degree there exists a polarization on the surface such that absolute stability with respect to it is equivalent to the stability of the restriction to the generic fibre. For degree 0, the result is no longer true, but if we consider semistability instead of stability we can adapt O'Grady's proof to show the following: Lemma 3.11.…”

Section: Corollary 34 Let G Be In D( X) the First Chern Charactersmentioning

confidence: 99%

Stable sheaves on elliptic fibrations

Porras

2002

Journal of Geometry and Physics

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Abstract. Let X → B be an elliptic surface and M(a, b) the moduli space of torsion-free sheaves on X which are stable of relative degree zero with respect to a polarization of type aH + bµ, H being the section and µ the elliptic fibre (b ≫ 0).We characterize the open subscheme of M(a, b) which is isomorphic, via the relative Fourier-Mukai transform, with the relative compactified Simpson Jacobian of the family of those curves D ֒→ X which are flat over B. This generalizes and completes earlier constructions due to Friedman, Morgan and Witten. We also study the relative moduli scheme of torsion-free and semistable sheaves of rank n and degree zero on the fibres. The relative Fourier-Mukai transform induces an isomorphic between this relative moduli space and the relative n-th symmetric product of the fibration. These results are relevant in the study of the conjectural duality between F-theory and the heterotic string. Elliptic fibrations and relative Fourier-Mukai transform1.1. Introduction. Recently there has been a growing interest in the moduli spaces of stable vector bundles on elliptic fibrations. Aside from their mathematical importance, these moduli spaces provide a geometric background to the study of some recent developments in string theory, notably in connection with the conjectural duality between F-theory and heterotic string theory ([13] , [14], [4], [10]).In this paper we study such moduli spaces, dealing both with the case of relatively and absolutely stable sheaves.We only consider elliptic fibrations p : X → B with a section H and geometrically integral fibres.In the first part we consider the "dual" elliptic fibrationp : X → B ([5]) defined as the compactified relative Jacobian of X → B (actually, X turns out to be isomorphic with X) and we introduce the relative Fourier-Mukai transform and its properties. This allows for a nice description of the spectral cover construction. Given a sheaf F on X → B flat over B and fibrewise torsion-free and semistable of rank n and degree 0, we define its spectral cover C(F ) ֒→ X as the closed subscheme defined by the 0th Fitting ideal of the first Fourier-Mukai transform F . It is finite over B and generically of degree n. When B is a smooth curve, the spectral cover is actually flat of degree n and F is torsion-free and rank one over C(F ). Atiyah, Tu and Friedman-Morgan-Witten structure theorems for semistable sheaves of degree zero on an elliptic curve ([2], [23], [13]) play a fundamental role in this section. By the invertibility of the Fourier-Mukai

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Smooth Four-Manifolds and Complex Surfaces

Cited by 278 publications

References 0 publications

Testing Seiberg-Witten Solution

Testing Seiberg-Witten Solution

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Stable sheaves on elliptic fibrations

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