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474
The Symplectic Thom Conjecture
"... In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then use ..."
Abstract
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Cited by 54 (5 self)
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, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds. 1.
Holomorphic triangle invariants and the topology of symplectic four-manifolds
- Duke Math. J
"... This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, which leads to new ..."
Abstract
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Cited by 46 (5 self)
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to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads
RELATIVE SYMPLECTIC CAPS, 4-GENUS AND FIBERED KNOTS
"... Abstract. We prove relative versions of the symplectic capping theorem and suf-ficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex bound-ary and a symplectic surface Σ in X such that ∂ ..."
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such that ∂Σ is a transverse knot in ∂X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y, S) symplectically. As a consequence we obtain a relative version of the Symplectic Thom conjecture. We also prove a relative version
Braid group actions on derived categories of coherent sheaves
- DUKE MATH. J
, 2001
"... This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is ..."
Abstract
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Cited by 255 (8 self)
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This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results
Transversality in elliptic Morse theory for the symplectic action
, 1999
"... Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the definition of symplectic Floer homology and hence in the proof of the Arnold conjecture. There is currently no other referen ..."
Abstract
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Cited by 155 (11 self)
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Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the definition of symplectic Floer homology and hence in the proof of the Arnold conjecture. There is currently no other
Proof of Gradient Conjecture of R. Thom
"... Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x0 ont ..."
Abstract
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Cited by 2 (0 self)
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Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x0
On the First Integral Conjecture of René Thom
, 710
"... Abstract. More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word “g ..."
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“generically”, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom’s conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class C k with k ≥ d, where d
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