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28
Nahm's Transformation for Instantons
, 1988
"... We describe in mathematical detail the Nahm transformation which maps antiself dual connections on the fourtorus (S 1 ) 4 onto antiself dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyperKahle ..."
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Cited by 29 (0 self)
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We describe in mathematical detail the Nahm transformation which maps antiself dual connections on the fourtorus (S 1 ) 4 onto antiself dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyperKahler) isometry. CERNTH.5108/88 July 1988 1 Introduction This paper deals with "magic" properties of U(n) antiself dual (asd) connections A on a C n \Gamma bundle E over a fourtorus T 4 . The "witchcraft" starts by introducing a family of Dirac operators coupled to (E; A) parametrized by the dual torus T 4 . The family index turns out to be a bundle E ! T 4 (under a genericity assumption on A) and comes equipped with a natural connection A, which is again asd (Theorem 1.5). This is Nahm's transform. Doing it once more with ( E; A), we obtain ( E; A) and a unitary equivalence (E; A) ( E; A). In other words the square of Nahm's transform is the identity (...
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 ..."
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Cited by 25 (4 self)
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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.
Generic metrics, irreducible rankone PU(2) monopoles, and transversality
 Comm. Anal. Geom
"... Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved t ..."
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Cited by 18 (7 self)
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Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved transversality using an
Diffeomorphism of simply connected algebraic surfaces
, 2004
"... In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a c ..."
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Cited by 11 (9 self)
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In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where ( by M. Freedman’s theorem) the topological type is completely determined by the numerical invariants of the surface. We hope that the methods of proof may turn out to be quite useful to show diffeomorphism and indeed symplectic equivalence for many important classes of algebraic surfaces and symplectic 4manifolds.
EMBEDDED SURFACES AND ALMOST COMPLEX STRUCTURES
, 1998
"... Abstract. In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4manifold X to be pseudoholomorphic with respect to an almost complex structure on X. In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not ..."
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Cited by 5 (1 self)
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Abstract. In this paper, we prove necessary and sufficient conditions for a smooth surface in a smooth 4manifold X to be pseudoholomorphic with respect to an almost complex structure on X. In particular, this provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.