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.A pseudo-holomorphic curve is a 2-submanifold embedded in a 2n-dimensional symplectic manifold X via a map which is holomorphic with respectto a Riemann surface structure on the domain and an almost-complex structureJ on X tamed by the symplectic form.Recent results by Taubes 48 , 49 , 50 have uncovered a deep relation be-tween these and the Seiberg Witten invariants in the case of four-dimensionalsymplectic manifolds.We shall not attempt here to present any of the details of Taubes' work, sincethe technical di culties involved are well beyond the purpose of this book.Werecommend the interested reader to approach this topic by rst reading thedetailed research announcement 48 and the survey paper 24 , and then thesubstantial part of the work 49 , 50.What we do in this section is just abrief exposition of the result, and some digressions on the theme of pseudoholomorphic curves.In our context, we consider the case of a 4-dimensional compact connectedsymplectic manifold without boundary.Given a homology class A 2 H2 X; Z ,Zlet H A be the space of J-holomorphic curves that realise the class A.Lemma 8.15 For a generic choice of a tamed almost-complex structure J, thespace H A is a smooth even dimensional manifold of dimension2dH = ,c1 K PD A + PD A ; X ;where K is the canonical line bundle de ned by the symplectic structure andPD A is the Poincar e dual of the class A.Notice that the canonical class c1 K often simply denoted K is well de-ned independently of the choice of the almost complex structure J, since theset J of !-tamed almost complex structures is contractible.This result is proven with the same technique we have used in computing thedimension of the Seiberg Witten moduli space: in fact the equation describingthe J-holomorphic condition linearises to a Fredholm operator.dHWhen dH 0, given a set of distinct points in X, we take H to be2the subspace of H A whose points are the curves that contain all of the pointsin.Theorem 8.16 For a generic choice of J and of the points in X, H is acompact zero dimensional manifold endowed with a canonical orientation.116 Some details can be found in the rst chapter of 36.This allows us tode ne invariants as follows.De nition 8.17 We de ne the Gromov invariant to be a map: H2 X; Z ! ZZ Zwhich is the sum with orientation of the points in H A , if dH 0; the sum ofpoints of H A with orientation, if dH =0; and zero by de nition when dH 0.The value of is independent of the choice of a generic set of points and ofthe quasi-complex structure J.The result announced in Taubes' paper 48 isthe following.Theorem 8.18 Given a Spinc structure s on a compact symplectic 4-manifoldX, with L = det S+ , we haveNs PD c1 L :The strategy of the proof is to extend to other Spinc-structures the asymp-totic technique introduced in 45 to compute the Seiberg Witten invariant of asymplectic manifold with respect to the canonical Spinc-structure.In this process, the asymptotic method shows how to associate a pseudoholomorphic curve to a solution of the Seiberg Witten equations, in the limitwhen the perturbation parameter satis es r !1.A di erent argument, whichis brie y sketched in 48 , provides a converse construction of a solution to theequations, given a J-holomorphic curve.A combined use of these constructionswould complete the proof of the theorem.A more precise way of stating the theorem is given in 49.Theorem 8.19 Let X; ! be a compact symplectic four-manifold with b+ 1.2If the Seiberg Witten invariant is non-zero Ns =0 for a certain Spinc-structure,6then there exists a J-holomorphic curve C which need not be connected and canhave multiplicities C = imiCi such thati the class C 2 H2 X; Z is the Poincar e dual of the class of the lineZbundle L,C = PD c1 L ;ii the intersection numbers satisfyK Ci Ci Ci;where K is the class of the canonical line bundle;iii the multiplicities mi satisfy mi =1 unless the component Ci is a torusof self intersection zero.117 This version of the theorem leads to a clear geometric interpretation of theresult.It is known in fact that in the case of K surfaces there are twoahlerequivalent ways of thinking of holomorphic curves: either as parametrised curvesfrom a Riemann surface that satisfy the Cauchy-Riemann equations or as thezero set of a holomorphic section of a holomorphic line bundle.The rst methodextends to symplectic geometry via the notion of J-holomorphic curves, whilethere is no available notion of holomorphic bundle" and holomorphic section".Away of thinking of Taubes' result as I rst learnt from lectures of S.Bauer andD [ Pobierz całość w formacie PDF ]
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.A pseudo-holomorphic curve is a 2-submanifold embedded in a 2n-dimensional symplectic manifold X via a map which is holomorphic with respectto a Riemann surface structure on the domain and an almost-complex structureJ on X tamed by the symplectic form.Recent results by Taubes 48 , 49 , 50 have uncovered a deep relation be-tween these and the Seiberg Witten invariants in the case of four-dimensionalsymplectic manifolds.We shall not attempt here to present any of the details of Taubes' work, sincethe technical di culties involved are well beyond the purpose of this book.Werecommend the interested reader to approach this topic by rst reading thedetailed research announcement 48 and the survey paper 24 , and then thesubstantial part of the work 49 , 50.What we do in this section is just abrief exposition of the result, and some digressions on the theme of pseudoholomorphic curves.In our context, we consider the case of a 4-dimensional compact connectedsymplectic manifold without boundary.Given a homology class A 2 H2 X; Z ,Zlet H A be the space of J-holomorphic curves that realise the class A.Lemma 8.15 For a generic choice of a tamed almost-complex structure J, thespace H A is a smooth even dimensional manifold of dimension2dH = ,c1 K PD A + PD A ; X ;where K is the canonical line bundle de ned by the symplectic structure andPD A is the Poincar e dual of the class A.Notice that the canonical class c1 K often simply denoted K is well de-ned independently of the choice of the almost complex structure J, since theset J of !-tamed almost complex structures is contractible.This result is proven with the same technique we have used in computing thedimension of the Seiberg Witten moduli space: in fact the equation describingthe J-holomorphic condition linearises to a Fredholm operator.dHWhen dH 0, given a set of distinct points in X, we take H to be2the subspace of H A whose points are the curves that contain all of the pointsin.Theorem 8.16 For a generic choice of J and of the points in X, H is acompact zero dimensional manifold endowed with a canonical orientation.116 Some details can be found in the rst chapter of 36.This allows us tode ne invariants as follows.De nition 8.17 We de ne the Gromov invariant to be a map: H2 X; Z ! ZZ Zwhich is the sum with orientation of the points in H A , if dH 0; the sum ofpoints of H A with orientation, if dH =0; and zero by de nition when dH 0.The value of is independent of the choice of a generic set of points and ofthe quasi-complex structure J.The result announced in Taubes' paper 48 isthe following.Theorem 8.18 Given a Spinc structure s on a compact symplectic 4-manifoldX, with L = det S+ , we haveNs PD c1 L :The strategy of the proof is to extend to other Spinc-structures the asymp-totic technique introduced in 45 to compute the Seiberg Witten invariant of asymplectic manifold with respect to the canonical Spinc-structure.In this process, the asymptotic method shows how to associate a pseudoholomorphic curve to a solution of the Seiberg Witten equations, in the limitwhen the perturbation parameter satis es r !1.A di erent argument, whichis brie y sketched in 48 , provides a converse construction of a solution to theequations, given a J-holomorphic curve.A combined use of these constructionswould complete the proof of the theorem.A more precise way of stating the theorem is given in 49.Theorem 8.19 Let X; ! be a compact symplectic four-manifold with b+ 1.2If the Seiberg Witten invariant is non-zero Ns =0 for a certain Spinc-structure,6then there exists a J-holomorphic curve C which need not be connected and canhave multiplicities C = imiCi such thati the class C 2 H2 X; Z is the Poincar e dual of the class of the lineZbundle L,C = PD c1 L ;ii the intersection numbers satisfyK Ci Ci Ci;where K is the class of the canonical line bundle;iii the multiplicities mi satisfy mi =1 unless the component Ci is a torusof self intersection zero.117 This version of the theorem leads to a clear geometric interpretation of theresult.It is known in fact that in the case of K surfaces there are twoahlerequivalent ways of thinking of holomorphic curves: either as parametrised curvesfrom a Riemann surface that satisfy the Cauchy-Riemann equations or as thezero set of a holomorphic section of a holomorphic line bundle.The rst methodextends to symplectic geometry via the notion of J-holomorphic curves, whilethere is no available notion of holomorphic bundle" and holomorphic section".Away of thinking of Taubes' result as I rst learnt from lectures of S.Bauer andD [ Pobierz całość w formacie PDF ]