# LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

Size: px
Start display at page:

Download "LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS"

Transcription

1 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be an ω-compatible almost complex structure. Let g J (, ) ω(, J ) be the corresponding hermitian metric (i.e. J-invariant Riemannian metric) on M. Let (Σ, j) be a Riemann surface (not necessarily compact) with complex structure j. A smooth map u : Σ M is called a (J, j)-holomorphic map (or simply a J-holomorphic map) if du j = J du, or equivalently, J (u) 1 (du + J du j) = 0. 2 The equation J (u) = 0 is a first order, non-linear equation of Cauchy-Riemann type. We give a description of it in a local coordinate system. Let z 0 Σ be any point and let p = u(z 0 ) M be its image in M under u. Supppose s + it is a local holomorphic coordinate centered at z 0 and φ : U R 2n is a local chart centered at p M. Set φ u = (u 1,, u 2n ) T. Then J (u) = 1 2 (( su j ) + J(u 1,, u 2n )( t u j ))ds (( tu j ) J(u 1,, u 2n )( s u j ))dt, and J (u) = 0 is equivalent to ( s u j ) + J(u 1,, u 2n )( t u j ) = 0. If J is integrable and (u 1,, u 2n ) is coming from a local holomorphic coordinate system (z 1,, z n ) with z j = u j +iu j+n, j = 1,, n, then J(u 1,, u 2n ) is constant in u 1,, u 2n and equals the matrix ( ) 0 I J 0 = I 0 where I denotes the n n identity matrix. Cauchy-Riemann equations In this case, J (u) = 0 becomes the s u j t u j+n = 0, s u j+n + s u j = 0, j = 1,, n. Hence when J is integrable, J-holomorphic maps are simply the usual holomorphic maps. On the other hand, it is easy to see that for a general J, the linearization of the non-linear equation J (u) = 0 is a zero-th order perturbation of the Cauchy-Riemann equations. Local properties. We shall next list several relevant local analytical properties of J-holomorphic maps. 1

2 2 WEIMIN CHEN, UMASS, SPRING 07 Let u, v : Σ M be two smooth maps and let z 0 Σ be a point. We say that u, v agree to the infinite order at z 0 if u(z 0 ) = v(z 0 ) = p 0, and there is a local chart centered at p 0, φ : U R 2n, such that all partial derivatives of the R 2n -valued function φ u φ v vanish at z 0. Proposition 1.1. (Unique continuation). If u, v : Σ M are two J-holomorphic maps which agree to the infinite order at a point z 0 Σ, then u v in the connetced component of Σ which contains z 0. Let u : Σ M be a J-holomorphic map. A point z Σ is called a critical point if du(z) = 0. Correspondingly the image u(z) M is called a critical value. We remark that u is locally an embedding at any point which is not a critical point. To see this, we suppose du(z) 0 for some z Σ. Let u(z) = p and let s + it be a local holomorphic coordinate centered at z. Then du(z) 0 means that either s u(z) T p M or t u(z) T p M is non-zero. But u is J-holomorphic so that s u + J(u) t u = 0, which implies that both s u(z), t u(z) T p M are non-zero. Hence u is locally an embedding near z. Lemma 1.2. A critical point of a non-constant J-holomorphic map is isolated. In particular, a non-constant J-holomorphic map from a compact Riemann surface has only finitely many critical points. Lemma 1.3. Let Ω C be an open neighborhood of 0 C and let u, v : Ω M be J-holomorphic maps such that u(0) = v(0), du(0) 0. Moreover, assume that there exist sequences z n, w n Ω such that u(z n ) = v(w n ), lim z n = lim w n = 0, w n 0. n n Then there exists a holomorphic function φ : B ɛ (0) Ω defined in some neighborhood of 0 C such that φ(0) = 0 and v = u φ. Lemmas 1.2 and 1.3 have the following consequence. Corollary 1.4. Let u : Σ M be a non-constant J-holomorphic map from a compact Riemann surface. Then there exists a compact Riemann surface Σ and a non-constant J-holomorphic map v : Σ M such that in the complement of finitely many points, v is an embedding onto its image. Moreover, there exists a biholomorphism or branched covering map φ : Σ Σ such that u = v φ. The map v in the above corollary is called simple and the map u is called multiply covered if deg(φ) > 1. The image C Im v is called a J-holomorphic curve in M, and the map v : Σ M is called a parametrization of C. We call C a rational J-holomorphic curve if Σ = S 2. Let u : Σ M be a smooth map, where Σ is given a complex structure j, M is given a J J (M, ω). We denote by g J the associated hermitian metric on M. In

3 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS 3 order to define the energy of the map u, we fix a Kähler metric h on Σ, and with h and g J, the norm du is well-defined. We define the energy of u to be E(u) du 2 dvol Σ. Σ An important fact about E(u) is that even though the energy density du 2 may depend on the choice of the Kähler metric h on Σ, the energy E(u) depends only on the complex structure j, i.e., E(u) is invariant under comformal transformations on the domain of u. The following energy identity can be easily derived E(u) = J (u) 2 dvol Σ + u ω, Σ which has the following important consequence. (This is where the closedness of ω plays a real role.) Proposition 1.5. J-holomorphic maps are the absolute minima of the energy functional E(u) amongst the smooth maps u which carry a fixed homology class in M. In particular, J-holomorphic maps are harmonic maps, and the energy of a J-holomorphic map depends only on the homology class it carries, and a J-holomorphic map must be constant if it carries a trivial homology class. Finally, we give the following important local analytical property of J-holomorphic maps. Theorem 1.6. (Removal of singularities) Let D C be the unit disc containing 0 and let u : D \ {0} M be a J-holomorphic map such that E(u) <. Then u may be extended to a J-holomorphic map û : D M with û D\{0} = u. Next we consider the moduli space of J-holomorphic maps. For simplicity, we shall assume Σ = S 2. In this case, the complex structure j is unique, and the group of biholomorphisms of Σ is the group of Möbius trnsformations G = PSL(2, C): z az + b, a, b, c, d C, ad bc = 1. cz + d Fix a non-zero homology class 0 A H 2 (M; Z). We consider the space of J- holomorphic maps M(A, J) = {u : S 2 M u is J-holomorphic and u [S 2 ] = A}, and the subspace of M(A, J) consisting of simple J-holomorphic maps M (A, J) = {u : S 2 M u is J-holomorphic and simple, and u [S 2 ] = A}. Note that the group G = PSL(2, C) acts on M(A, J) via reparametrization φ u = u φ 1, φ G, u M(A, J), which is free when restricted on the subspace M (A, J). We denote the quotient space by M(A, J) and M (A, J) respectively. Note that M (A, J) is exactly the space of J-holomorphic curves C such that the homology class of C is A. We remark that Σ

4 4 WEIMIN CHEN, UMASS, SPRING 07 when A is a primitive class, i.e., A is not an integral multiple of another integral class, M(A, J) = M (A, J). Compactness. One of the fundamental issues concerning the moduli spaces is compactness. Note that the group G = PSL(2, C) acts freely on M (A, J) and G is not a compact group. Hence the moduli space of J-holomorphic maps M(A, J) and M (A, J) can not be compact, and one could best hope that the quotient spaces M(A, J) and M (A, J) are compact. However, this is also not true in general, as illustrated in the following example. Example 1.7. Consider a family of holomorphic curves of degree 2 in CP 2 parametrized by 0 λ C C λ = {[z 0, z 1, z 2 ] λz 2 0 = z 1 z 2 } M (2[CP 1 ], J 0 ). Here [CP 1 ] H 2 (CP 2 ; Z) is the class of a line, and J 0 is the complex structure of CP 2. As λ 0, C λ converges to a union of two lines C 0 = {[z 0, z 1, z 2 ] z 1 z 2 = 0} = {[z 0, 0, z 2 ]} {[z 0, z 1, 0]} which intersect transversely at [1, 0, 0]. It is known that C 0 can not be the image of a holomorphic map u : S 2 CP 2, hence C 0 does not lie in M(2[CP 1 ], J 0 ). This shows that both M(2[CP 1 ], J 0 ) and M (2[CP 1 ], J 0 ) are non-compact. The phenomenon illustrated in the above example is called bubbling, i.e., during the limiting process as λ 0, the holomorphic curves C λ split off a (non-constant) J- holomorphic 2-sphere which carries strictly less energy than the original curves. The bubbling phenomenon is the primary cause of non-compactness of moduli space of J-holomorphic curves, and when Σ = S 2 as what we currently consider, it is the only cause. In other words, if there is no bubbling, the space M(A, J) is compact. Next we give a simple criterion which ensures compactness. Recall that a homology class B H 2 (M; Z) is called spherical if it may be represented by a map from S 2 into M. Suppose the symplectic manifold (M, ω) contains no spherical classes B such that 0 < ω(b) < ω(a). Such a condition has two consequences: (1) every element u M(A, J) is simple because otherwise the image of u represents a spherical class B satisfying 0 < ω(b) < ω(a), this gives M (A, J) = M(A, J), (2) there is no bubbling for elements in M(A, J) because a split-off J-holomorphic 2-sphere would represent a spherical class B satisfying 0 < ω(b) < ω(a). This gives rise to the following simple version of the Gromov Compactness Theorem. Theorem 1.8. (Gromov). Suppose there are no spherical classes B such that 0 < ω(b) < ω(a). Then for any compact subset W J (M, ω) (given with C -topology), J W M(A, J) is compact with respect to the C -topology. The full version of the Gromov Compactness Theorem states that the moduli space of J-holomorphic curves carrying a fixed homology class can be suitably compactified. This is where the closedness of ω plays a real role, cf. Proposition 1.5.

5 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS 5 Fredholm theory. Finally, we discuss the Fredholm theory of J-holomorphic maps, which allows us to analyze the topological structure of the moduli spaces. Fix a sufficiently large integer l > 0, we consider the Banach manifold B {u : S 2 M u is a C l -map and u [S 2 ] = A} and the Banach bundle E B, where the fiber over u B is E u {v v is a C l 1 -section of Hom(T S 2, u T M) S 2 such that v j = J v}. The Banach bundle E B has a natural smooth section s : B E defined by s : u (u, J (u)). By the elliptic regularity of the equation J (u) = 0, any C l -solution is automatically a smooth solution, so that the moduli space of J-holomorphic maps M(A, J) is simply the zero loci of s, i.e., s 1 (zero-section) = M(A, J). A crucial fact is that s : B E is a Fredholm section, which means that the linearization of J (u) for each u B, D u : T u B E u, is a Fredholm operator between the Banach spaces. This has the following implication on the topological structure of the moduli space M(A, J). For any open subset U M(A, J), if D u : T u B E u is onto for any u U, then U is a canonically oriented, finite dimensional smooth manifold whose dimension is given by the index of D u, which can be computed via the Atiyah- Singer index theorem in the following formula Index D u = 2n(1 g S 2) + 2c 1 (T M) A. Here 2n = dim M and g S 2 = 0 is the genus of S 2. Such a J is called regular (with respect to U). (We remark that the same holds true if one allows J to vary in an oriented finite dimensional space.) When J is integrable, the operator D u : T u B E u is simply the -operator : Ω 0 (CP 1, V ) Ω 0,1 (CP 1, V ) where V = u T M is a holomorphic vector bundle over CP 1. The cokernel of D u is simply the Dolbeault cohomology group H 0,1 (CP1, V ), which by Kodaira-Serre duality is isomorphic to the space of holomorphic sections of V K. Here V is the dual of V and K is the canonical bundle of CP 1. The following lemma follows immediately from vanishing theorems of holomorphic vector bundles. Lemma 1.9. Suppose J is integrable and V = u T M CP 1 is a holomorphic vector bundle of non-negative curvature tensor. Then D u is onto. In general, using the Sard-Smale theorem one has Theorem There exists an open, dense subset J reg (A) J (M, ω) of second Bair category such that for any J J reg (A), J is regular with respect to M (A, J), so that M (A, J) is a smooth manifold of dimension dim M + 2c 1 (T M) A.

6 6 WEIMIN CHEN, UMASS, SPRING 07 Moreover, for any J 1, J 2 J reg (A), there exists a path J t J (M, ω) connecting J 1, J 2 such that t M (A, J t ) is an oriented smooth manifold with boundary which is the disjoint union of M (A, J 1 ) and M (A, J 2 ). 2. The non-squeezing theorem and Gromov invariant As one of the first applications of J-holomorphic curve theory, we describe the proof of the following non-squeezing theorem, where B 2n (R) denotes the closed ball of radius R in R 2n which is equipped with the standard symplectic structure ω 0. Theorem 2.1. (Gromov, 1985). B 2 (r) R 2n 2 if r < 1. There exist no symplectic embeddings B 2n (1) Proof. Suppose to the contrary, there exists an symplectic embedding ψ : B 2n (1) B 2 (r) R 2n 2 for some r < 1. Fix any ɛ > 0, we consider B 2 (r) as a subset of S 2 which is given a symplectic form σ with total area πr 2 + ɛ. On the other hand, since ψ(b 2n (1)) is compact, its projection into the R 2n 2 factor is contained in an open ball of radius λ centered at the origin. Let T 2n 2 be the torus which is R 2n 2 modulo the lattice {(x 1,, x 2n ) λ x j Z}, which inherits a natural symplectic form ω 0. We set M = S 2 T 2n 2, which is given with the product symplectic structure ω = σ ω 0. With this understood, note that there is a symplectic embedding ψ : (B 2n (1), ω 0 ) (M, ω). We set p 0 = ψ(0) where 0 B 2n (1) is the origin. Lemma 2.2. For any J J (M, ω), there exists a ratinal J-holomorphic curve C which contains p 0 and carries the homology class [S 2 {pt}]. Assuming Lemma 2.2 momentarily, the proof of Theorem 2.1 goes as follows. Note that there is a J J (M, ω) such that the pull-back almost complex structure ψ J is the standard complex structure J 0 on B 2n (1). Let C be the rational J-holomorphic curve which contains p 0 and carries the homology class [S 2 {pt}]. We set C ψ 1 (C) B 2n (1). Then C is a holomorphic curve in B 2n (1) containing the origin. Particularly, C is a minimal surface, and by the theory of minimal surfaces, the area of C is at least the area of the flat plane contained in B 2n (1), which equals π. This gives rise to the following inequalities π Area(C ) = ω 0 = ω ω = σ = πr 2 + ɛ. C ψ(c ) C S 2 Let ɛ 0, we obtain π πr 2, which contradicts the assumption r < 1. This proves the non-squeezing theorem. The basic idea behind the proof of Lemma 2.2 is the so-called Gromov invariant, which is the number of rational J-holomorphic curves (counted with signs) for a given J, that carries a given homology class and satisfies a certain topological constraint. (Such a count of J-holomorphic curves is supposed to be independent of the choice of J.) Lemma 2.2 basically says that the Gromov invariant which counts the

7 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS 7 number of rational curves carrying a homology class [S 2 {pt}] and passing through a given point in M is non-zero. We shall next explain how to define such a Gromov invarint in the current context, and explain why the Gromov invariant is non-zero. To this end, we set A = [S 2 {pt}] H 2 (M; Z). Since ω = σ ω 0 is a product symplectic structure, c 1 (T M) = c 1 (T S 2 ) + c 1 (T 2n 2 ), so that c 1 (T M) A = c 1 (T S 2 ) A = 2. By Theorem 1.10, there is an open, dense subset of second Bair category J reg (A) J (M, ω), such that for any J J reg (A), the space M (A, J) is an oriented smooth manifold of dimension dim M + 2c 1 (T M) A = 2n + 4. In the present case, since A is a generator of H 2 (M, Z) = Z, there are no spherical classes B such that 0 < ω(b) < ω(a), so that by Theorem 1.8, M(A, J) = M (A, J), and the quotient space M(A, J) is compact, which is an oriented smooth manifold of dimension dim M(A, J) = dim M (A, J) dim PSL(2, C) = 2n = 2n 2. Denote PSL(2, C) by G, and set M(A, J) G S 2 (M(A, J) S 2 )/G where G acts on M(A, J) S 2 via φ (u, z) = (u φ 1, φ(z)). Then M(A, J) G S 2 is a compact, oriented smooth manifold of dimension 2n, which is a S 2 -bundle over M(A, J). The evaluation map ev : M(A, J) G S 2 M, [(u, z)] u(z) is a smooth map between two compact, oriented smooth manifolds of the same dimension. The degree of ev, which is the image of the fundamental class of M(A, J) G S 2 under ev : H 2n (M(A, J) G S 2 ; Z) H 2n (M; Z) = Z, can be geometrically interpreted as a count with signs of the points in the pre-image ev 1 (p) for any generic point p M. On the other hand, M(A, J) G S 2 as a S 2 -bundle over M(A, J) may be regarded as the space of rational J-holomorphic curves C M(A, J) with a marked point z S 2 in the de-singularization of C. Thus the degree of ev is a count with signs of the number of rational J-holomorphic curves with a marked point, which carry the homology class A and pass through a given generic point p M at the marked point. The Gromov invariant involved in the current problem is defined to be the degree of the evaluation map ev : M(A, J) G S 2 M. Note that the Gromov invariant is independent of the choice of J J reg (A). This is because by Theorem 1.10, for any J 1, J 2 J reg (A), there exists a path J t J (M, ω) connecting J 1, J 2 such that t M(A, J t ) is an oriented smooth manifold with boundary which is the disjoint union of M(A, J 1 ) and M(A, J 2 ). It follows that t M(A, J t ) G S 2 is a cobordism between M(A, J 1 ) G S 2 and M(A, J 2 ) G S 2, hence the degree of ev is the same for J 1, J 2. This shows that the Gromov invariant is independent of the choice of J J reg (A). In order to show that the Gromov invariant is non-zero, we consider a special J J reg (A). Let j, J 0 be the complex structure on S 2 and T 2n 2 respectively, and let J = j J 0 be the product which lies in J (M, ω).

8 8 WEIMIN CHEN, UMASS, SPRING 07 For any u M(A, J), since J = j J 0, the map pr u : S 2 T 2n 2, where pr : M T 2n 2 is the projection, is J 0 -holomorphic. But pr u carries a trivial homology class, hence by Proposition 1.5, pr u is a constant map. This shows that any u M(A, J) has the form u : z (φ(z), x) for some φ G = PSL(2, C) and x T 2n 2. There are two consequences of this fact: (1) For any u M(A, J), u T M is isomorphic as a holomorphic vector bundle to T S 2 E where E is a trivial bundle of rank n 1. By Lemma 1.9, D u is onto for any u M(A, J), so that J J reg (A). (2) The correspondence u (φ, x) gives an identification of M(A, J) with G T 2n 2, and hence M(A, J) with T 2n 2 and M(A, J) G S 2 with S 2 T 2n 2 = M. It follows that the evaluation map ev : M(A, J) G S 2 M is a diffeomorphism, and the degree of ev is ±1. This proves that the Gromov invariant is non-zero. Proof of Lemma 2.2. Note that the non-vanishing of Gromov invariant only implies immediately that for any J J reg (A), and for any generic point p M, there exists a J-holomorphic curve C M(A, J) such that p C. This is different from the claim in Lemma 2.2 that in fact such a J-holomorphic curve exists for any J J (M, ω) and any point p M (in particular, p 0 M). To get around of this, we use the Gromov Compactness Theorem, Theorem 1.8. We pick a sequence of J n J reg (A), since J reg (A) is dense in J (M, ω), which converges to J J (M, ω) in C -topology, and we pick a sequence of generic points p n converging to p 0 M, such that for each n, there exists a J n -holomorphic curve C n such that p n C n. By Theorem 1.8, a subsequence of {C n } converges to a C M(A, J) such that p 0 = lim n p n C. This proves Lemma J-holomorphic curves in dimension 4 The J-holomorphic curve theory in dimension 4 is particularly more powerful because there are additional tools which allow one to analyse the singularies of a J- holomorphic curve. On the other hand, the existence of certain types of J-holomorphic curves actually can be derived from the underlying differential topology of the symplectic 4-manifold, due to the deep analytical work of Cliff Taubes. Let (M, J) be an almost complex 4-manifold, and let C M be a J-holomorphic curve parametrized by a simple J-holomorphic map u : Σ M. The following theorem gives a criterion, amongst other things, for the embeddedness of C. Theorem 3.1. (Adjunction Inequality). Let g Σ be the genus of Σ. Then the inequality 1 2 (C2 c 1 (T M) C) + 1 g Σ holds with equality if and only if C is embedded. In particular, a rational J-holomorphic curve must be embedded if it is homologous to an embedded rational J-holomorphic curve. This explains why the singular curve C 0 in Example 1.7 can not be the image of a holomorphic map u : S 2 CP 2.

9 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS 9 Example 3.2. (Algebraic curves in CP 2 ). For notations we denote by [CP 1 ] the generator of H 2 (CP 2 ; Z) which is the class of a line. Using Poincaré duality, we identify H 2 (CP 2 ; Z) with H 2 (CP 2 ; Z). We first consider non-singular algebraic curves in CP 2. Let C be a line in CP 2. Then C 2 = 1, and 1 2 (C2 c 1 (T CP 2 ) C) + 1 = 0, which implies that c 1 (T CP 2 ) = 3 [CP 1 ]. Now let C d be any non-singular algebraic curve of degree d. Then Cd 2 = d2 and c 1 (T CP 2 ) C d = 3d. This gives rise to the following genus formula for C d : genus (C d ) = 1 2 (C2 d c 1(T CP 2 ) C d ) + 1 = 1 (d 1)(d 2). 2 Next we consider a singular algebraic curve, the cusp curve C 0 = {[z 0, z 1, z 2 ] CP 2 z 3 1 = z 0 z 2 2}. C 0 is of degree 3 and has a cusp singularity at [1, 0, 0]. adjunction inequality for C 0 is 1 2 (C2 0 c 1 (T CP 2 ) C 0 ) + 1 = 1 2 (32 3 3) + 1 = 1. The left-hand side of the Since C 0 is singular, C 0 can only be parametrized by a holomorphic map from a genus zero Riemann surface, i.e., S 2, so C 0 belongs to M (3[CP 1 ], J 0 ). On the other hand, C 0 is the limit of a family of non-singular cubic curves (λ 0) C λ = {[z 0, z 1, z 2 ] CP 2 z 3 1 = z 0 z λz 3 0} as λ 0. We remark that this also represents a certain kind of non-compactness phenomenon in the Gromov Compactness Theorem. As an application of Theorem 3.1, we prove the following non-existence result. Proposition 3.3. For a generic almost complex structure J, there exist no rational J-holomorphic curves C such that C 2 2, and there exist at most embedded rational J-holomorphic curves C with C 2 = 1. Proof. Suppose C is a rational J-holomorphic curve such that C 2 2. Then the adjunction inequality implies that c 1 (T M) C C = 0. On the other hand, for a generic almost complex structure J (cf. Theorem 1.10), the space M ([C], J) is a smooth manifold of dimension 4 + 2c 1 (T M) C 4. Since G = PSL(2, C) is 6-dimensional and acts on M ([C], J) freely if it is non-empty, we see that M ([C], J) must be at least 6-dimensional. This proves that for a generic almost complex structure, there exist no rational curves C with C 2 2. The proof for the case of C 2 = 1 is similar and we leave the details to the reader.

10 10 WEIMIN CHEN, UMASS, SPRING 07 The above proposition shows that in a symplectic 4-manifold, the only interesting rational J-holomorphic curves are those with non-negative self-intersection. Because if J is taken generic, the only rational J-holomorphic curves with negative self-intersection are the embedded ones with self-intersection 1. By the symplectic neighborhood theorem, the symplectic 4-manifold can be symplectically blown down along these ( 1)-curves, and the resulting symplectic 4-manifold does not have any rational J-holomorphic curves with negative self-intersection for a generic J. A symplectic 4-manifold is called minimal if it contains no embedded symplectic 2-spheres with self-intersection 1 (i.e., it can not be symplectically blown down). The following theorem is useful in analysing the intersection of two distinct J- holomorphic curves. Theorem 3.4. (Positivity of Intersection) Let C, C be two distinct J-holomorphic curves in a compact almost complex 4-manifold. Then the intersection of C, C consists of at most finitely many points. Moreover, the intersection product C C = p C C k p where k p Z +, and k p = 1 if and only if both C, C are embedded near p and the intersection at p is transverse. In particular, C C 0, and if C C = 0, then C, C are disjoint. If C C = 1, then C, C intersect at exactly one point and the intersection must be transverse. Suppose C is an embedded rational J-holomorphic curve with C 2 = 0, and suppose C is another rational J-holomorphic curve which is homologous to C. Then on the one hand, the adjunction inequality implies C must also be embedded, and on the other hand, the positivity of intersection implies that C, C must be disjoint. Thus if the moduli space of such rational curves has a positive dimension, they may be used to fill up the whole manifold. In order to do this, we need the following regularity criterion for J. Lemma 3.5. Suppose C is an immersed rational J-holomorphic curve in an almost complex 4-manifold (M, J) such that c 1 (T M) C > 0. Then for any simple J-holomorphic map u : S 2 M parametrizing C, the linearization D u of J (u) = 0 is onto. We combine these tools to give a proof of the following structural theorem of symplectic 4-manifolds which contain an embedded rational curve of self-intersection 0. Theorem 3.6. Let (M, ω) be a symplectic 4-manifold which contains an embedded symplectic 2-sphere C with C 2 = 0. Suppose that M contains no spherical classes B such that 0 < ω(b) < ω(c). Then M must be diffeomorphic to a S 2 -bundle over a surface. Proof. Pick a J J (M, ω) such that C is J-holomorphic. Then by the adjunction inequality every element in M ([C], J) is embedded, and since c 1 (T M) C = C = 2 > 0,

11 LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS 11 by Lemma 3.5, the space M ([C], J) is an oriented smooth manifold of dimension dim M + 2c 1 (T M) C = = 8. Consequently, M ([C], J) is an oriented surface, which is compact by Theorem 1.8. With this understood, M is diffeomorphic to the S 2 -bundle over M ([C], J) via where G = PSL(2, C). ev : M ([C], J) G S 2 M, [(u, z)] u(z), Suppose in the above theorem, M contains another embedded symplectic 2-sphere C such that C C = 0, which intersects with C transversely and positively at a single point. Then one can arrange J J (M, ω) such that both C, C are J- holomorphic. Suppose furthermore that there are also no spherical classes B such that 0 < ω(b) < ω(c ), then M ([C ], J) is precisely the space of J-holomorphic sections of M ([C], J) G S 2 under ev. It is easily seen that in this case there is a diffeomorphism ψ : S 2 S 2 M such that the 2-spheres ψ(s 2 {pt}) and ψ({pt} S 2 ) are symplectic in M. This fact was exploited in the proof of the following theorem. Theorem 3.7. (Gromov-Taubes). For any symplectic structure ω on CP 2, there is a diffeomorphism ψ : CP 2 CP 2 such that ψ ω is a multiple of the standard Fubini- Study form ω 0. The proof of Theorem 3.7 consists of two steps. Step 1, which is due to Taubes and uses his deep work on Seiberg-Witten theory of symplectic 4-manifolds, asserts that there exists an embedded symplectic 2-sphere C with C 2 = 1. The complement CP 2 \ ν(c) of a regular neighborhood of C is a homotopy 4-ball W with W = S 3 (Van-Kampen plus Mayer-Vietoris). By the symplectic neighborhood theorem, the symplectic form ω equals the standard symplectic form on R 4 in a regular neighborhood of W, which is identified with {x R 4 δ 0 ɛ < x 2 δ 0 } for some δ 0 and ɛ > 0. Step 2: (Gromov). There exists a symplectomorphism B 4 (δ 0 ) W which is identity near the boundaries. The proof of Step 2 goes as follows. Pick a large enough polydisc D 2 D 2 R 4 which contains B 4 (δ 0 ), and embedded D 2 D 2 into S 2 S 2 via some embedding D 2 S 2. Then one removes B 4 (δ 0 ) from S 2 S 2 and then glues back W. Call the resulting symplectic 4-manifold M. Apply the remarks following Theorem 3.6 to M (for details see [1], pages ). References [1] D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, Colloquium Publications 52, AMS, 2004.

### LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

### J-curves and the classification of rational and ruled symplectic 4-manifolds

J-curves and the classification of rational and ruled symplectic 4-manifolds François Lalonde Université du Québec à Montréal (flalonde@math.uqam.ca) Dusa McDuff State University of New York at Stony Brook

### J-holomorphic curves in symplectic geometry

J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely

### Nodal symplectic spheres in CP 2 with positive self intersection

Nodal symplectic spheres in CP 2 with positive self intersection Jean-François BARRAUD barraud@picard.ups-tlse.fr Abstract 1 : Let ω be the canonical Kähler structure on CP 2 We prove that any ω-symplectic

### Topics in Geometry: Mirror Symmetry

MIT OpenCourseWare http://ocw.mit.edu 8.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

### Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

### The topology of symplectic four-manifolds

The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form

### Symplectic geometry on moduli spaces of J-holomorphic curves

Symplectic geometry on moduli spaces of J-holomorphic curves J. Coffey, L. Kessler, and Á. Pelayo Abstract Let M, ω be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form ω SiΣ on

### L19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1

L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness,

### LECTURE 2: SYMPLECTIC VECTOR BUNDLES

LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over

### LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

### Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions

### COUNT OF GENUS ZERO J-HOLOMORPHIC CURVES IN DIMENSIONS FOUR AND SIX arxiv: v5 [math.sg] 17 May 2013

COUNT OF GENUS ZERO J-HOLOMORPHIC CURVES IN DIMENSIONS FOUR AND SIX arxiv:0906.5472v5 [math.sg] 17 May 2013 AHMET BEYAZ Abstract. In this note, genus zero Gromov-Witten invariants are reviewed and then

### LECTURE 4: SYMPLECTIC GROUP ACTIONS

LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is

### Donaldson Invariants and Moduli of Yang-Mills Instantons

Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants

### Lecture XI: The non-kähler world

Lecture XI: The non-kähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The non-kähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of

### Determinant lines and determinant line bundles

CHAPTER Determinant lines and determinant line bundles This appendix is an exposition of G. Segal s work sketched in [?] on determinant line bundles over the moduli spaces of Riemann surfaces with parametrized

### Symplectic 4-manifolds, singular plane curves, and isotopy problems

Symplectic 4-manifolds, singular plane curves, and isotopy problems Denis AUROUX Massachusetts Inst. of Technology and Ecole Polytechnique Symplectic manifolds A symplectic structure on a smooth manifold

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

### Lagrangian Intersection Floer Homology (sketch) Chris Gerig

Lagrangian Intersection Floer Homology (sketch) 9-15-2011 Chris Gerig Recall that a symplectic 2n-manifold (M, ω) is a smooth manifold with a closed nondegenerate 2- form, i.e. ω(x, y) = ω(y, x) and dω

### We have the following immediate corollary. 1

1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E

### DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 1, January 1998, Pages 305 310 S 000-9939(98)04001-5 DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES TERRY FULLER (Communicated

### Scalar curvature and the Thurston norm

Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,

### Complex manifolds, Kahler metrics, differential and harmonic forms

Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on

### MORSE HOMOLOGY. Contents. Manifolds are closed. Fields are Z/2.

MORSE HOMOLOGY STUDENT GEOMETRY AND TOPOLOGY SEMINAR FEBRUARY 26, 2015 MORGAN WEILER 1. Morse Functions 1 Morse Lemma 3 Existence 3 Genericness 4 Topology 4 2. The Morse Chain Complex 4 Generators 5 Differential

### PICARD S THEOREM STEFAN FRIEDL

PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A

### Smooth s-cobordisms of elliptic 3-manifolds

University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2006 Smooth s-cobordisms of elliptic 3-manifolds

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### PART 2: PSEUDO-HOLOMORPHIC CURVES

PART 2: PSEUDO-HOLOMORPHIC CURVES WEIMIN CHEN, UMASS, FALL 09 Contents 1. Properties of J-holomorphic curves 1 1.1. Basic definitions 1 1.2. Unique continuation and critical points 5 1.3. Simple curves

### Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable

### Part II. Riemann Surfaces. Year

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

### Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

### Notes by Maksim Maydanskiy.

SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### Handlebody Decomposition of a Manifold

Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody

### LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

### The Canonical Sheaf. Stefano Filipazzi. September 14, 2015

The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over

### LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

### Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

### 0.1 Complex Analogues 1

0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson

### LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

### THE UNIFORMISATION THEOREM OF RIEMANN SURFACES

THE UNIFORISATION THEORE OF RIEANN SURFACES 1. What is the aim of this seminar? Recall that a compact oriented surface is a g -holed object. (Classification of surfaces.) It can be obtained through a 4g

### Lecture Fish 3. Joel W. Fish. July 4, 2015

Lecture Fish 3 Joel W. Fish July 4, 2015 Contents 1 LECTURE 2 1.1 Recap:................................. 2 1.2 M-polyfolds with boundary..................... 2 1.3 An Implicit Function Theorem...................

### 1 Hermitian symmetric spaces: examples and basic properties

Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

### X G X by the rule x x g

18. Maps between Riemann surfaces: II Note that there is one further way we can reverse all of this. Suppose that X instead of Y is a Riemann surface. Can we put a Riemann surface structure on Y such that

### Hofer s Proof of the Weinstein Conjecture for Overtwisted Contact Structures Julian Chaidez

Hofer s Proof of the Weinstein Conjecture for Overtwisted Contact Structures Julian Chaidez 1 Introduction In this paper, we will be discussing a question about contact manifolds, so let s start by defining

### FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS

FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.

### THE QUANTUM CONNECTION

THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

### Lecture III: Neighbourhoods

Lecture III: Neighbourhoods Jonathan Evans 7th October 2010 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 1 / 18 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 2 / 18 In

### 1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

### Linear connections on Lie groups

Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

### STANDARD SURFACES AND NODAL CURVES IN SYMPLECTIC 4-MANIFOLDS

STANDARD SURFACES AND NODAL CURVES IN SYMPLECTIC 4-MANIFOLDS MICHAEL USHER Abstract. Continuing the program of [DS] and [U1], we introduce refinements of the Donaldson-Smith standard surface count which

### 30 Surfaces and nondegenerate symmetric bilinear forms

80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

### Morse theory and stable pairs

Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

### CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

### INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

### Bordism and the Pontryagin-Thom Theorem

Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

### Embedded contact homology and its applications

Embedded contact homology and its applications Michael Hutchings Department of Mathematics University of California, Berkeley International Congress of Mathematicians, Hyderabad August 26, 2010 arxiv:1003.3209

### THE CANONICAL PENCILS ON HORIKAWA SURFACES

THE CANONICAL PENCILS ON HORIKAWA SURFACES DENIS AUROUX Abstract. We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that

### Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.

Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections

### Broken pencils and four-manifold invariants. Tim Perutz (Cambridge)

Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### From symplectic deformation to isotopy

From symplectic deformation to isotopy Dusa McDuff State University of New York at Stony Brook (dusa@math.sunysb.edu) May 15, 1996, revised Aug 4, 1997 Abstract Let X be an oriented 4-manifold which does

### THE HODGE DECOMPOSITION

THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional

### Patrick Iglesias-Zemmour

Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

### Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

### ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1

ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a

### A Crash Course of Floer Homology for Lagrangian Intersections

A Crash Course of Floer Homology for Lagrangian Intersections Manabu AKAHO Department of Mathematics Tokyo Metropolitan University akaho@math.metro-u.ac.jp 1 Introduction There are several kinds of Floer

### Cobordant differentiable manifolds

Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated

### An introduction to cobordism

An introduction to cobordism Martin Vito Cruz 30 April 2004 1 Introduction Cobordism theory is the study of manifolds modulo the cobordism relation: two manifolds are considered the same if their disjoint

### Lecture 2. Smooth functions and maps

Lecture 2. Smooth functions and maps 2.1 Definition of smooth maps Given a differentiable manifold, all questions of differentiability are to be reduced to questions about functions between Euclidean spaces,

### QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

### Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds

Self-intersections of Closed Parametrized Minimal Surfaces in Generic Riemannian Manifolds John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: moore@math.ucsb.edu

### Let X be a topological space. We want it to look locally like C. So we make the following definition.

February 17, 2010 1 Riemann surfaces 1.1 Definitions and examples Let X be a topological space. We want it to look locally like C. So we make the following definition. Definition 1. A complex chart on

### Tools from Lebesgue integration

Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

### ON NEARLY SEMIFREE CIRCLE ACTIONS

ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)

### Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

### PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

### MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

### AN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS

AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### Math 6510 Homework 10

2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained

### MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract

### Lagrangian knottedness and unknottedness in rational surfaces

agrangian knottedness and unknottedness in rational surfaces Outline: agrangian knottedness Symplectic geometry of complex projective varieties, D 5, agrangian spheres and Dehn twists agrangian unknottedness

### Stable bundles with small c 2 over 2-dimensional complex tori

Stable bundles with small c 2 over 2-dimensional complex tori Matei Toma Universität Osnabrück, Fachbereich Mathematik/Informatik, 49069 Osnabrück, Germany and Institute of Mathematics of the Romanian

### B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

### Rohlin s Invariant and 4-dimensional Gauge Theory. Daniel Ruberman Nikolai Saveliev

Rohlin s Invariant and 4-dimensional Gauge Theory Daniel Ruberman Nikolai Saveliev 1 Overall theme: relation between Rohlin-type invariants and gauge theory, especially in dimension 4. Background: Recall

### Intermediate Jacobians and Abel-Jacobi Maps

Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Introduction Let X be a smooth projective complex variety. Introduction Let X be a smooth projective complex variety. Intermediate

### UNCERTAINTY PRINCIPLE, NON-SQUEEZING THEOREM AND THE SYMPLECTIC RIGIDITY LECTURE FOR THE 1995 DAEWOO WORKSHOP, CHUNGWON, KOREA

UNCERTAINTY PRINCIPLE, NON-SQUEEZING THEOREM AND THE SYMPLECTIC RIGIDITY LECTURE FOR THE 1995 DAEWOO WORKSHOP, CHUNGWON, KOREA Yong-Geun Oh, Department of Mathematics University of Wisconsin Madison, WI

### Lecture I: Overview and motivation

Lecture I: Overview and motivation Jonathan Evans 23rd September 2010 Jonathan Evans () Lecture I: Overview and motivation 23rd September 2010 1 / 31 Difficulty of exercises is denoted by card suits in

### Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli