LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear
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Arxiv:1704.04175V1 [Math.DG] 13 Apr 2017 Umnfls[ Submanifolds Mty[ Ometry Eerhr Ncmlxgoer.I at H Loihshe Algorithms to the Aim Fact, the in with Geometry
SAGEMATH EXPERIMENTS IN DIFFERENTIAL AND COMPLEX GEOMETRY DANIELE ANGELLA Abstract. This note summarizes the talk by the author at the workshop “Geometry and Computer Science” held in Pescara in February 2017. We present how SageMath can help in research in Complex and Differential Geometry, with two simple applications, which are not intended to be original. We consider two "classification problems" on quotients of Lie groups, namely, "computing cohomological invariants" [AFR15, LUV14], and "classifying special geo- metric structures" [ABP17], and we set the problems to be solved with SageMath [S+09]. Introduction Complex Geometry is the study of manifolds locally modelled on the linear complex space Cn. A natural way to construct compact complex manifolds is to study the projective geometry n n+1 n of CP = C 0 /C×. In fact, analytic submanifolds of CP are equivalent to algebraic submanifolds [GAGA\ { }]. On the one side, this means that both algebraic and analytic techniques are available for their study. On the other side, this means also that this class of manifolds is quite restrictive. In particular, they do not suffice to describe some Theoretical Physics models e.g. [Str86]. Since [Thu76], new examples of complex non-projective, even non-Kähler manifolds have been investigated, and many different constructions have been proposed. One would study classes of manifolds whose geometry is, in some sense, combinatorically or alge- braically described, in order to perform explicit computations. In this sense, great interest has been deserved to homogeneous spaces of nilpotent Lie groups, say nilmanifolds, whose ge- ometry [Bel00, FG04] and cohomology [Nom54] is encoded in the Lie algebra. -
Symplectic Linear Algebra Let V Be a Real Vector Space. Definition 1.1. A
INTRODUCTION TO SYMPLECTIC TOPOLOGY C. VITERBO 1. Lecture 1: Symplectic linear algebra Let V be a real vector space. Definition 1.1. A symplectic form on V is a skew-symmetric bilinear nondegen- erate form: (1) ω(x, y)= ω(y, x) (= ω(x, x) = 0); (2) x, y such− that ω(x,⇒ y) = 0. ∀ ∃ % For a general 2-form ω on a vector space, V , we denote ker(ω) the subspace given by ker(ω)= v V w Vω(v, w)=0 { ∈ |∀ ∈ } The second condition implies that ker(ω) reduces to zero, so when ω is symplectic, there are no “preferred directions” in V . There are special types of subspaces in symplectic manifolds. For a vector sub- space F , we denote by F ω = v V w F,ω(v, w) = 0 { ∈ |∀ ∈ } From Grassmann’s formula it follows that dim(F ω)=codim(F ) = dim(V ) dim(F ) Also we have − Proposition 1.2. (F ω)ω = F (F + F )ω = F ω F ω 1 2 1 ∩ 2 Definition 1.3. A subspace F of V,ω is ω isotropic if F F ( ω F = 0); • coisotropic if F⊂ω F⇐⇒ | • Lagrangian if it is⊂ maximal isotropic. • Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1,L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism ϕ : V V such that ϕ∗ω = ω and ϕ(L )=L . -
On the Complexification of the Classical Geometries And
On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical -
Symplectic Geometry
Vorlesung aus dem Sommersemester 2013 Symplectic Geometry Prof. Dr. Fabian Ziltener TEXed by Viktor Kleen & Florian Stecker Contents 1 From Classical Mechanics to Symplectic Topology2 1.1 Introducing Symplectic Forms.........................2 1.2 Hamiltonian mechanics.............................2 1.3 Overview over some topics of this course...................3 1.4 Overview over some current questions....................4 2 Linear (pre-)symplectic geometry6 2.1 (Pre-)symplectic vector spaces and linear (pre-)symplectic maps......6 2.2 (Co-)isotropic and Lagrangian subspaces and the linear Weinstein Theorem 13 2.3 Linear symplectic reduction.......................... 16 2.4 Linear complex structures and the symplectic linear group......... 17 3 Symplectic Manifolds 26 3.1 Definition, Examples, Contangent bundle.................. 26 3.2 Classical Mechanics............................... 28 3.3 Symplectomorphisms, Hamiltonian diffeomorphisms and Poisson brackets 35 3.4 Darboux’ Theorem and Moser isotopy.................... 40 3.5 Symplectic, (co-)isotropic and Lagrangian submanifolds.......... 44 3.6 Symplectic and Hamiltonian Lie group actions, momentum maps, Marsden– Weinstein quotients............................... 48 3.7 Physical motivation: Symmetries of mechanical systems and Noether’s principle..................................... 52 3.8 Symplectic Quotients.............................. 53 1 From Classical Mechanics to Symplectic Topology 1.1 Introducing Symplectic Forms Definition 1.1. A symplectic form is a non–degenerate and closed 2–form on a manifold. Non–degeneracy means that if ωx(v, w) = 0 for all w ∈ TxM, then v = 0. Closed means that dω = 0. Reminder. A differential k–form on a manifold M is a family of skew–symmetric k– linear maps ×k (ωx : TxM → R)x∈M , which depends smoothly on x. 2n Definition 1.2. We define the standard symplectic form ω0 on R as follows: We 2n 2n 2n 2n identify the tangent space TxR with the vector space R . -
Complex Cobordism and Almost Complex Fillings of Contact Manifolds
MSci Project in Mathematics COMPLEX COBORDISM AND ALMOST COMPLEX FILLINGS OF CONTACT MANIFOLDS November 2, 2016 Naomi L. Kraushar supervised by Dr C Wendl University College London Abstract An important problem in contact and symplectic topology is the question of which contact manifolds are symplectically fillable, in other words, which contact manifolds are the boundaries of symplectic manifolds, such that the symplectic structure is consistent, in some sense, with the given contact struc- ture on the boundary. The homotopy data on the tangent bundles involved in this question is finding an almost complex filling of almost contact manifolds. It is known that such fillings exist, so that there are no obstructions on the tangent bundles to the existence of symplectic fillings of contact manifolds; however, so far a formal proof of this fact has not been written down. In this paper, we prove this statement. We use cobordism theory to deal with the stable part of the homotopy obstruction, and then use obstruction theory, and a variant on surgery theory known as contact surgery, to deal with the unstable part of the obstruction. Contents 1 Introduction 2 2 Vector spaces and vector bundles 4 2.1 Complex vector spaces . .4 2.2 Symplectic vector spaces . .7 2.3 Vector bundles . 13 3 Contact manifolds 19 3.1 Contact manifolds . 19 3.2 Submanifolds of contact manifolds . 23 3.3 Complex, almost complex, and stably complex manifolds . 25 4 Universal bundles and classifying spaces 30 4.1 Universal bundles . 30 4.2 Universal bundles for O(n) and U(n).............. 33 4.3 Stable vector bundles . -
Symmetries of the Space of Linear Symplectic Connections
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 002, 30 pages Symmetries of the Space of Linear Symplectic Connections Daniel J.F. FOX Departamento de Matem´aticas del Area´ Industrial, Escuela T´ecnica Superior de Ingenier´ıa y Dise~noIndustrial, Universidad Polit´ecnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain E-mail: [email protected] Received June 30, 2016, in final form January 07, 2017; Published online January 10, 2017 https://doi.org/10.3842/SIGMA.2017.002 Abstract. There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen{Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle poly- nomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term. -
Almost Complex Structures and Obstruction Theory
ALMOST COMPLEX STRUCTURES AND OBSTRUCTION THEORY MICHAEL ALBANESE Abstract. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018. Let X be a CW complex and Y a simply connected space. Last time we discussed the obstruction to extending a map f : X(n) ! Y to a map X(n+1) ! Y ; recall that X(k) denotes the k-skeleton of X. n+1 There is an obstruction o(f) 2 C (X; πn(Y )) which vanishes if and only if f can be extended to (n+1) n+1 X . Moreover, o(f) is a cocycle and [o(f)] 2 H (X; πn(Y )) vanishes if and only if fjX(n−1) can be extended to X(n+1); that is, f may need to be redefined on the n-cells. Obstructions to lifting a map p Given a fibration F ! E −! B and a map f : X ! B, when can f be lifted to a map g : X ! E? If X = B and f = idB, then we are asking when p has a section. For convenience, we will only consider the case where F and B are simply connected, from which it follows that E is simply connected. For a more general statement, see Theorem 7.37 of [2]. Suppose g has been defined on X(n). Let en+1 be an n-cell and α : Sn ! X(n) its attaching map, then p ◦ g ◦ α : Sn ! B is equal to f ◦ α and is nullhomotopic (as f extends over the (n + 1)-cell). -
1 Symplectic Vector Spaces
Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U. -
Complex Structures and 2 X 2 Matrices
COMPLEX STRUCTURES AND 2 2 MATRICES × Simon Salamon University of Warwick, 14 May 2004 Part I Complex structures on R4 Part II SKT structures on a 6-manifold Part I The standard complex structure A (linear) complex structure on R4 is simply a linear map J: R4 R4 satisfying J 2 = 1. For example, ! − 0 1 0 1 0 J0 = 0 − 1 0 0 1 B 1 0 C B − C @ A In terms of a basis (dx1; dx2; dx3; dx4) of R4 , Jdx1 = dx2; Jdx3 = dx4 − − The choices of sign ensure that, setting z1 = x1 + ix2; z2 = x3 + ix4; the elements dz1; dz2 C4 satisfy 2 Jdz1 = J(dx1 + idx2) = i(dx1 + idx2) = idz1 Jdz2 = idz2: 2 Spaces of complex structures Any complex structure J is conjugate to J0 , so the set of complex structures is 1 = A− J0A : A GL(4; R) C f 2 g GL(4; R) = ; ∼ GL(2; C) a manifold of real dim 8, the same dim as GL(2; C). + Taking account of orientation, = − . C C t C Theorem + S2 . C ' Proof. Any J + has a polar decomposition 2 C 1 1 T 1 1 J = SO = O− S− = (O S− O)( O− ): − − 1 1 Thus, O = O− = P − J P defines an element − 0 SO(4) U(2)P = S2: 2 U(2) ∼ 3 Deformations and 2 2 matrices × A complex structure J is determined by its i-eigenspace 1;0 4 E = Λ = v C : Jv = iv ; f 2 g i on E since E E = C4 and J = ⊕ i on E: − For example, E = dz1; dz2 and E = dz1; dz2 . -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
1. Overview Over the Basic Notions in Symplectic Geometry Definition 1.1
AN INTRODUCTION TO SYMPLECTIC GEOMETRY MAIK PICKL 1. Overview over the basic notions in symplectic geometry Definition 1.1. Let V be a m-dimensional R vector space and let Ω : V × V ! R be a bilinear map. We say Ω is skew-symmetric if Ω(u; v) = −Ω(v; u), for all u; v 2 V . Theorem 1.1. Let Ω be a skew-symmetric bilinear map on V . Then there is a basis u1; : : : ; uk; e1 : : : ; en; f1; : : : ; fn of V s.t. Ω(ui; v) = 0; 8i and 8v 2 V Ω(ei; ej) = 0 = Ω(fi; fj); 8i; j Ω(ei; fj) = δij; 8i; j: In matrix notation with respect to this basis we have 00 0 0 1 T Ω(u; v) = u @0 0 idA v: 0 −id 0 In particular we have dim(V ) = m = k + 2n. Proof. The proof is a skew-symmetric version of the Gram-Schmidt process. First let U := fu 2 V jΩ(u; v) = 0 for all v 2 V g; and choose a basis u1; : : : ; uk for U. Now choose a complementary space W s.t. V = U ⊕ W . Let e1 2 W , then there is a f1 2 W s.t. Ω(e1; f1) 6= 0 and we can assume that Ω(e1; f1) = 1. Let W1 = span(e1; f1) Ω W1 = fw 2 W jΩ(w; v) = 0 for all v 2 W1g: Ω Ω We claim that W = W1 ⊕ W1 : Suppose that v = ae1 + bf1 2 W1 \ W1 . We get that 0 = Ω(v; e1) = −b 0 = Ω(v; f1) = a and therefore v = 0. -
Symplectic and Kaehler Geometry
Chapter 3 Symplectic and Kaehler Geometry Lecture 12 Today: Symplectic geometry and Kaehler geometry, the linear aspects anyway. Symplectic Geometry Let V be an n dimensional vector space over R, B : V V R a bilineare form on V . × → Definition. B is alternating if B(v, w) = B(w, v). Denote by Alt2(V ) the space of all alternating bilinear forms on V . − Definition. Take any B Alt(V ), U a subspace of V . Then we can define the orthogonal complement by ∈ U ⊥ = v V, B(u, v) = 0, u U { ∈ ∀ ∈ } Definition. B is non-degenerate if V ⊥ = 0 . { } Theorem. If B is non-degenerate then dim V is even. Moeover, there exists a basis e1, . , en, f1, . , fn of V such that B(ei, en) = B(fi, fj ) = 0 and B(ei, fj ) = δij Definition. B is non-degenerate if and only if the pair (V, B) is a symplectic vector space. Then ei’s and fj ’s are called a Darboux basis of V . Let B be non-degenerate and U a vector subspace of V Remark: dim U ⊥ = 2n dim V and we have the following 3 scenarios. − 1. U isotropic U ⊥ U. This implies that dim U n ⇔ ⊃ ≤ 2. U Lagrangian U ⊥ = U. This implies dim U = n. ⇔ 3. U symplectic U ⊥ U = . This implies that U ⊥ is symplectic and B and B ⊥ are non-degenerate. ⇔ ∩ ∅ |U |U Let V = V m be a vector space over R we have 2 2 Alt (V ) ∼= Λ (V ∗) is a canonical identification. Let v1, . , vm be a basis of v, then 2 1 Alt (V ) B B(v , v )v∗ v∗ ∋ 7→ 2 i j i ∧ j 2 2 � and the inverse Λ (V ∗) ω B Alt (V ) is given by ∋ 7→ ω ∈ B(v, w) = iW (iV ω) Suppose m = 2n.