The manifold of compatible almost complex structures and. Trends in complex analysis, differential geometry, and. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost complex lie algebroids. Mathematical surveys and monographs volume 98 moment maps, cobordisms, and hamiltonian group actions victor guillemin viktor ginzburg yael karshon. Next the almost hermitian lie algebroids and some related structures on the associated complex lie algebroid are studied. Geometric quantization, complex structures and the. Several results concerning the computations of dolbeault and bottchern cohomologies on nilmanifolds are summarized, allowing readers to study explicit examples. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. Pdf dynamics equations with almost complex structures on. Almost complex structures on a manifold and its exotic copy. The manifold of compatible almost complex structures and geometric quantization. Stable complex and spincstructures in this book, gmanifolds are often equipped with a stable complex structure or a spinc structure. Methods of the theory of lie groups allow one to construct a wide class of homogeneous spaces having an invariant almostcomplex structure both integrable and nonintegrable and to classify invariant almostcomplex structures under different assumptions see. Recent title s in thi s series american mathematical society.
Kahler manifolds lie at the intersection of differential geometry, complex analysis, algebraic geometry, pdes, microlocal analysis, probability, and topology. Applications of harmonic analysis in quantum statistical mechanics and quantum field theory by n. For example, the similarity between the heisenberg. Konstantin, bulgaria, 36 september 2002 bulgaria international workshop on complex structures, vector fields 6th 2002 varna, dimiev s.
Almost complex structures and geometric quantization 3 where tdx is the todd class of x. Other readers will always be interested in your opinion of the books youve read. A bynow standard example is geometric quantization is the construction of the segalbargmann space for cn using geometric quantization with a. Quantization of the laplacian operator on vector bundles, i. Almost complex and complex structures pure mathematics. What are the best almost complex structures on the 6sphere. The global theory of geometric structures is quite complicated and is far from being completed.
A real vector space with a complex structure can be given the structure of a. Nunes october 27, 2004 abstract it is shown that the heat operator in the hall coherent state trans. The dolbeault complex may be replaced with the rolled up version 1. Jan 21, 2015 the almost complex lie algebroids over smooth manifolds are considered in the paper. David borthwick, alejandro uribe, almost complex structures and geometric quantization arxiv. This article studies a class of connections defined on a symplectic manifold with a lagrangian foliation that model certain aspects of local differential geometry. Robinson 403 motion of curves by crystalline curvature, including triple junctions and boundary points. The second chapter is devoted to constructions of twistor spaces over riemannian manifolds. The paperback of the geometric quantization in action. Complex time evolution in geometric quantization jos e mourao t ecnico lisboa, u lisboa seminar. Design and analysis of tall and complex structures 1st edition. Almost complex structures that model nonlinear geometries. Of course, while lagrangian mechanics, to a certain extent, retains the standard differential geometric form, hamiltonian mechanics alters it significantly enough to give it a special name, symplectic geometry.
Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology. Almost complex manifolds almost complex structures. Now suppose that xhas merely an almost complex structure and lis a hermitian line bundle with compatible hermitian connection. What are the best almostcomplex structures on the 6sphere. We may clinearly extend i to a1 x, and thus obtain a decomposition a1 x a 1,0. Geometric quantization oxford mathematical monographs. Pdf adapted complex structures and geometric quantization. For a more gauge theoryoriented treatment the text dk90 of donaldson and kronheimer is more suitable. Duistermaat at the meeting of february 26, 2001 in this paper, a footnote to the theory in gu and bu, we give an. We define three cohomologies on an almost complex manifold m, j, defined using the nijenhuislie derivations induced from the almost complex structure j and its nijenhuis tensor n, regarded as vectorvalued forms on m. Methods of the theory of lie groups allow one to construct a wide class of homogeneous spaces having an invariant almost complex structure both integrable and nonintegrable and to classify invariant almost complex structures under different assumptions see. Manifolds endowed with almost complex structures, or with other special structures such as, for example, symplectic, generalized complex, etc.
Complex manifolds and deformation of complex structures. A complex manifold is a smooth manifold with complex valued coordinate functions that depend on one another holomorphically on coordinate patch intersections. A previous paper by one of us 64 referred to the symplectic creed. Zhang nankai university \two themes in geometric quantization i would like to survey my two works related to geometric quantization and symplectic reduction. Aadepartment of mathematics, university of western ontario, abmathematics department, university of michigan. Faculty of sciences and technology, university of macau october, 16, work in collaboration with will kirwin u k oln and joao p.
We prove the analog of kodaira vanishing for the spinc dirac. Geometric quantization and dynamical constructions on the space of kahler metrics, 2008. Design and analysis of tall and complex structures 1st. Almost complex structures and geometric quantization. Deformation theory of global structure was initiated by kodairaspencer. The book 29 treats further topics in symplectic geometry. We study two quantization schemes for compact symplectic manifolds with almost complex structures. Tour start here for a quick overview of the site help center detailed answers to any questions you might have meta discuss the workings and policies of this site. In quantum mechanics, the motion of the harmonic oscillator is described by a complex valued wave function. Geometric quantization and related complex geometry. We obtain asymptotics of the curvature of a natural connection in these bundles. Hitchins projectively flat connection and the moduli. The rst is my joint work with youliang tian on the analytic approach of the guilleminsternberg geometric quantization conjecture. The books of souriau 1970 and simms and woodhouse 1976 present the theory of geometric quantization and its relationship to quantum mech anics.
Almost complex structures have important applications in symplectic geometry the concept is due to charles ehresmann and heinz. A complex structure on a real vector space v is a linear endomorphism j of v such that j2. Konstantin, bulgaria, 36 september 2002 bulgaria international workshop on complex structures, vector. The purpose of the present book is to complement the preceding ones by including new developments of the theory and emphasizing the computations leading to results in quantum mechanics. Trends in complex analysis, differential geometry, and mathematical physics. We prove the analog of kodaira vanishing for the spinc dirac operator. A remark on almost complex quantization in symplectic. Roughly speaking, a complex structure on v enable us to \multiply p 1 on v and thus convert v into a complex vector space.
In this chapter, we study the asymptotic expansion of the bergman kernel associated to modified dirac operators and renormalized bochner laplacians on symplectic manifolds. Algebraic embeddings of complex and almost complex. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Manifolds endowed with almostcomplex structures, or with other special structures such as, for example, symplectic, generalizedcomplex, etc. Almost complex structures have important applications in symplectic geometry the concept is due to charles ehresmann and heinz hopf in. In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Geometric mechanics is classical mechanics formulated in the language of modern differential geometry. This site is like a library, use search box in the widget to get ebook that you want. Canonical almost complex structure on symplectic manifold. Calabivessiniti and kuranishi studied deformations of complex structures based on hodge theory. It has given valuable and lasting insights into the relationship between classical and quantum systems, and continues to be a popular research topic. Dynamics equations with almost complex structures on contact 5manifolds that the existence of the stack of microdifferential modules on an arbitrary contact manifold, although he.
Recall that a complex structure on a real vector space v is automorphism j. The manifold of compatible almost complex structures and geometric quantization article in communications in mathematical physics 2742. Click download or read online button to get lectures on symplectic manifolds book now. Geometry of complex and algebraic manifolds unifies riemannian geometry with modern complex analysis, as well as with algebra and number theory. Twistor spaces and harmonic maps institute of physics. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. The geometric approach to quantization was introduced by konstant and souriau more than 20 years ago.
We prove the analog of kodaira vanishing for the spinc dirac operator, which shows that the index space of this operator provides an honest not virtual vector space semiclassically. We suppose that a compact lie group gacts on mpreserving e, such that the gorbits. Nicholas woodhouse, geometric quantization and quantum field theory in curved spacetimes, reports on mathematical physics 12. Almost cr quantization via the index of transversally elliptic dirac operators daniel sean fitzpatrick doctor of philosophy graduate department of mathematics university of toronto 2009 let mbe a smooth compact manifold equipped with a cooriented subbundle e. Almost complex structures and geometric quantization david borthwick and alejandro uribe abstract. Woodhouse gives a method of building quantum hilbert spaces from classical systems.
Purchase design and analysis of tall and complex structures 1st edition. The integrability condition allows to introduce local complex coordinates on m such that the transition functions between different patches are holomorphic. As a curiosity, note that two centuries ago the name symplectic geometry did. Almost cr quantization via the index of transversally. An integrable almost complex structure on a manifold. Almostcomplex structure encyclopedia of mathematics. A proof of the arnold conjecture by polyfold techniques, with p. We study a family of hermitian vector bundles on the space \\mathcalj\ of almost complex structures on m compatible with. One of these can be applied to distinguish nonisomorphic nonintegrable almost complex structures on m. The weak arnold conjecture is a lower bound on the number of periodic orbits of any periodic hamiltonian system in terms of the rank of homology of the underlying phase space. In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. Bergman kernels on symplectic manifolds springerlink. Calabi, weil, borel, matsushima and others studied deformations of geometric. Moment maps, cobordisms, and hamiltonian group actions.
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