Almost complex vector bundle pdf

For x a manifold we say that x has an almost complex structure respectively stable almost complex structure if. Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. Cr structures on open manifolds rutgers university. A complex vector bundle is a holomorphic vector bundle if x is a complex manifold and if the local. We discuss the hodge theory for hermitian bundles over compact almost complex manifolds in details and are able to show theorem 1.

Atiyahs classification of vector bundles on elliptic curves. We introduce connection and curvature of complex line bundle and prove that any curvature 2form for a complex line bundle is both closed an integral. Xis a complex topological vector bundle where the total space ehas been endowed with a complex structure making the projection. We will generalize the standard one to the new ones such that the induced 0, 2tensor on the tangent bundle using these structures and liouville 1form will be a riemannian metric.

Almost complex structures and geometric quantization. Pdf almost complex structures on spheres researchgate. Keywords riemannian manifolds, almost hermitian and almost hyper hermitian structures, tangent bundle 1. Roughly speaking, a complex structure on v enable us to \multiply p 1 on v and thus convert v into a complex vector space. Part of chapter 2, introducing ktheory, then proving bott periodicity in the complex case and adams theorem on the hopf invariant, with its famous applications to division algebras and parallelizability of spheres. Then one can hope to deform such a lift, which is itself an almost complex structure, to an integrable almost complex structure. Given two complex manifolds m1 and m2 a morphism f. The \holomorphic tangent bundle of a complex manifold recall that for any complex manifold x, there is a decomposition of the complexi ed tangent bundle. Symplectic linear group and linear complex structures15 4.

As the differential geometric counterpart to the stability, i introduced the concept of an einsteinhermitian vector bundle. A generalized almost contact structure on mis a sub bundle eof e1m c such that eis isotropic and e1m. In this paper, the standard almost complex structure on the tangent bunle of a riemannian manifold will be generalized. Old and new structures on the tangent bundle munteanu, marian ioan, 2007. Tm is simply what we call an almost complex structure on m. Let ebe a complex vector bundle with a pseudoholomrphic structure over a compact almost complex manifold x,j, then h0x,e is. They show that there is a bijection between bundle almost complex structures and the pseudoholomorphic structures on esee proposition 4. The only two vector bundles with base space a circle and onedimensional. Such a manifold mis called an almost complex manifold.

Let e be a smooth complex vector bundle over an almost complex manifold m, j of complex dimension n. A connection of type 1, 0 on e is a differential operator. Almost complex manifolds almost complex structures. Characterizing moisezon spaces by almost positive coherent. Some relationships between the geometry of the tangent bundle and the geometry of the riemannian base manifold henry, guillermo and keilhauer, guillermo, tokyo journal of mathematics, 2012. An almost complex structure on m is a fiberwise complex struc ture on the tangent bundle tm, i. Recall that a complex structure on a real vector space v is automorphism j. An almostcomplex manifold is a smooth real manifold j tm. Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has recently obtained new impetus from physicists interest in questions related to mirror symmetry. The real linear map jextends to a complex linear map jon this space, and since j2 id. On the geometry of tangent bundles induced by the isotropic. Show that if m has an almost complex structure then it must be evendimensional.

For example, gxw snx is weakly almost complex for all n but is almost complex only when n 3 or 7 1. V v is an almost complex structure, then dimrv is even and v admits, in a natural way, the structure of a complex vector space. The tangent bundle of an almost complex free loopspace morava, jack, homology, homotopy and applications, 2001. A bundle almost complex structure j on e is a special almost complex structure on the total space of the e, see 5,6. Recall that a smooth manifold m is said to be weakly almost complex if its tangent bundle xm is stably isomorphic to the realification of a complex vector bundle over m. The basic invariant of a complex vector bundle is a chern class. The linear algebra introduced above may be applied pointwise to the tangent. If x is a complex manifold, then the total space of the complex vector bundle t1,0x. By definition, a complex structure is a bundle map between a real vector bundle e and itself. Let e be a nef vector bundle on a compact complex connected.

It is well known that the tensor bundle of type 1, q of m admits an almost complex structure on the pure crosssection, if m admits an almost complex structure. This paper originates from the talk almost complex structures on. The dolbeault complex may be replaced with the rolled up version 1. Denote by a p, q m, e the space of smooth sections of the bundle of p, q forms valued in e. Geometry of twistor spaces claude lebrun simons workshop lecture, 73004 lecture notes by jill mcgowan. More generally, an almost complex structure on a real di. A real oriented vector bundle e is said to admit an almost complex structure if there is a complex vector bundle f such that. There is a bijection between bundle almost complex structures and. Then metrics always exist, and every sympletic vector bundle is a complex vector bundle and vice versa.

A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank saji, kentaro and tsuchida, asahi, rocky mountain journal of mathematics, 2019. We show that in finitedimensional nonlinear approximations, the best rterm approximant of a function f almost always exists over c but that the same is not true over r, i. Such a j is called an almost complex structure and makes the real tangent bundle into a complex vector bundle via declaring iv jv. The latter, viewed as a complex vector bundle using j. I am writing because i am extremely confused with the structure of complex vector bundles. If eis a complex vector bundle over mand, is a hermitian.

For p2m we consider the complexi ed tangent space t pm c, which is a complex vector space of dimension dimmand a real vector space of dimension 2dimm. Almost complex structure on a real vector space let v be a. General natural riemannian almost product and parahermitian structures on tangent bundles drutaromaniuc, simonaluiza, taiwanese journal of mathematics, 2012. If x is a complex manifold, then the total space of the complex vector bundle t1, 0x. A vector bundle has the local appearance of a product, but may have nontrivial global topology. This assertion, while elementary, is fundamental in the arguments that follow, so we will prove it.

In this context h can be regarded as a jinvariant complex symmetric bilinear form on the complexi cation of the almost complex vector bundle tm, and. Okumura, on the almost complex structure of tangen bundles of riemannian spaces. The existence of an almost complex structure j on a manifold m is. Almost complex structures on connected sums of complex projective. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. This paper discussed vector bundle and complex line bundle,which is a special case of a. Show that the nontrivial vector bundle on s1 constructed above is not orientable.

Differential geometry of generalized lagrangian functions okubo, katsumi, journal of mathematics of kyoto university, 1991. The curvature fl of rl is an imaginary valued 2form on m. Note that an almost complex structure does not imply that the manifold admits a complex structure. Complex vector bundle vs holomorphic vector bundle vs almost. A complex manifold is thus in a canonical way an almost complex manifold. Almost hermitian structures on tangent bundles hiroyasu satoh abstract in this article, we consider the almost hermitian structure on tm induced by a pair of a metric and an a. Every positive vector bundle in the sense of grauert is almost positive. J, e of order 1 acting on the complex a, and satisfying the following two. We can find similar situation in mathematical analysis real and complex. Embeddings of almost hermitian manifold in almost hyper. An almostcomplex manifold is a smooth real manifold j tm jx ix m.

More generally, if xis any space, we can identify the homology groups e nxwith bordism groups of stably almost complex nmanifolds equipped with a map to x. Complex best rterm approximations almost always exist in. Moreover x is almost homogeneous, with automorphisms induced. We will focus on complex vector bundles, but for almost everything a real. Remarks on some almost hermitian structure on the tangent bundle, ii oguro, takashi and takashima, yoshitsugu, nihonkai mathematical journal, 2014. We now introduce the notion of semipositive vector bundles. Now suppose that xhas merely an almost complex structure and lis a hermitian line bundle with compatible hermitian connection. Curvature of a complex line bundle and hermitian line bundle. A bundle over a manifold is trivial if it is simply the cartesian product of.

Due to its interactions with various other fields differential. An almost complex manifold is a ck real 2nmanifold together with a ck endomorphism iof its tangent bundle i. They may also be expressed as moduli stacks of almost complex structures, see here. Since tx is not complex manifold, how to construct a holomorphic vector bundle. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Let f be a real vector space and let w be a complex. Analysis on almost complex manifolds almost complex manifolds suppose m. Any complex vector bundle over a paracompact space admits a hermitian metric. First, consider the integrability condition for almost complex structures on a. A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. Complex vector bundle vs holomorphic vector bundle vs. A complex vector bundle is a fiber bundle with fiber cn and structure group gln,c.

Holomorphic vector bundles northwestern university. Whenever a is a complex manifold, the complexified tangent bundle splits into type components. The linear algebra introduced above may be applied pointwise to the tangent bundle of m. The space of compatible almostcomplex structures on m.

Almost complex manifolds an almostcomplex manifold is a smooth real manifold m equipped with a smooth endomorphism. The main purpose of this paper is to investigate a similar problem for tensor bundles p tmq of type, p q, p 1. Introduction an almost complex structure on an even dimensional smooth manifold m is a smooth linear vector bundle map j on the tangent bundle of m satisfying j2. The pair m,j is called an almost complex manifold, and whenever j arises from 1. A generalized almost complex structure is a choice of a halfdimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. An almost complex structure on f is a complex isomorphism. Equivalently, each tangent space t pmis equipped with a complex structure j pwhich varies smoothly with respect to p. A nonzero vector bundle is called decomposable if it is the whitney sum of two nonzero vector bundles. An almost complex structure on v is a linear transformation j. Complex vector bundles and complex manifolds definition 1. This concept has been generalized to vector bundles and, more generally, coherent sheaves over algebraic manifolds by takemoto, bogomolov and gieseker. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. Contact manifolds and generalized complex structures. Almost complex structures and geometric quantization 3 where tdx is the todd class of x.

A vector bundle e over a complex space x is called semipositive if the following two conditions are satisfied. Vanishing theorems for the halfkernel of a dirac operator 5 suppose e is a cli. Ramanan no part of this book may be reproduced in any form by print, micro. Another question, when does a real manifold carry a complex structure so that it is indeed a complex manifold. As we have seen in the previous section, holomorphic vector bundles are complex vector bundles for which the local trivialization has been chosen in such a way that the transition functions are holomorphic. A complex vector bundle is a fiber bundle with fiber cn and structure group gln, c. A generalized almost contact structure on mis a subbundle eof e1m c such that eis isotropic and e1m. Complex algebraic geometry 5 is the clinear extension of f. Each j pmakes t pminto a complex vector space, and with j, tmbecomes a complex vector bundle. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. When m is regarded as a real manifold, tm is a real vector bundle of rank 2n. Strictly speaking, a vector bundle is an entire triple e. Lecture notes geometry of manifolds mathematics mit. It is important to note that, given a topological or smooth vector bundle e, holomorphicity.

Michele audin, symplectic and almost complex manifolds. More precisely, let ebe a complex vector bundle over an almost complex manifold x,j. Analysis on almost complex manifolds almost complex manifolds. The complex structure of m defines an almost complex structure j. A new class of almost complex structures on tangent bundle of. The complex projective spaces cpn form a fundamental class of k. Almost complex and hermitian vector spaces and vector bundles a let jbe an almost complex structure on a 2ndimensional real vector space v i. An almost complex structure j on m is a linear complex structure that is, a linear map which squares to.

In other words, we have a smooth tensor field j of degree 1, 1 such that. Given a lie group g, a principal g bundle over a space bcan be viewed as a parameterized family of spaces f x, each with a free, transitive action of gso in particular each f x is homeomorphic to g. Vector bundles and connections universiteit utrecht. An ordinary almost complex structure is a choice of a halfdimensional subspace of each fiber of the complexified tangent bundle tm. Let l be a hermitian line bundle endowed with a hermitian connection rl and let e be a hermitian vector bundle over m endowed with an hermitian connection re. A new class of almost complex structures on tangent bundle. Find materials for this course in the pages linked along the left. But one suspects that this may not be possible, in the same general way that. An almost complex structure on a 4dimensional oriented vector space is called orientation compatible if there is an oriented basis of the form e. By construction, trivial vector bundles are always orientable we can construct them from glkcocycles whose maps are constant, valued at the identity map.