That is to say that at a point x, ax is a reallinear map ax. Pdf download complex manifolds without potential theory free. Demailly, complex analytic and differential geometry pdf. Existence of a complex structure on the 6 dimensional sphere is proved in this paper. The differential geometrical methods of this theory were developed essentially under the. Almostcomplex and hypercomplex manifolds are considered in this paper from the point of view of complex analysis and potential theory. Potential obstructions to the possibility of patching are measured by. With an appendix on the geometry of characteristic classes universitext on free shipping on qualified orders.
X b of compact complex manifolds as a proper holomorphic submersion of complex manifolds. These notes grew out of a course called complex manifolds and hermitian differential. Finite energy pluripotential theory accommodates the variational theory of equations of complex mongeamp ere type arising in k ahler geometry. The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. Sidharth kshatriya under my guidance during the academic year 20062007. The authors intention is to introduce the reader in a simple way to the most important branches and methods in the theory of several complex variables.
Complex manifolds without potential theory 1995 differential equations on complex manifolds 1994 foliations by complex manifolds involving the complex hessian 1994 geometry and. If the complex manifold m may be covered by a bounded homogeneous domain in cn then it can be decomposed into a product of complex manifolds diffeomorphic to biquotient spaces r\gk where g is a noncompact simple lie group with a trivial center, k is a maximal connected compact lie subgroup of g and r is a subgroup of g acting in a free. A good introduction to the theory of complex manifolds, a subject that is far deeper than just smooth manifold theory with the word \smooth replaced by. Shiingshen chern, complex manifolds without potential theory springerverlag press, 1995 isbn 0387904220, isbn 3540904220. Mathematical methods in field theory and complex analytic varieties 1971. M we will denote by t x and t x the tangent and cotangent spaces respectively. This cited by count includes citations to the following articles in scholar. A concise course in complex analysis and riemann surfaces. Based on notes taken by james morrow from lectures given by kunihiko kodaira at stanford university in 19651966, the book gives the original proof of the kodaira embedding theorem, showing that the restricted class of kahler manifolds called hodge manifolds is. However, for functions u with dux 0, there is no natural definition of. These functions are defined classically by requiring that the restriction to each pseudoholomorphic curve is.
From holomorphic functions to complex manifolds a valuable addition to the literature. They are necessarily of even dimension, say 2n, and allow for a collection of charts u j,z j that are one to one maps of the corresponding u j to cn such that for every nonempty. It is known 7,9 that the model cp1 is equivalent to the o3 q model but the connection has not been made explicit in terms of o2 supersymmetry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, e ectively turning the. Pdf complex and hermitian structures on a vector space. The proof is based on reinterpreting a hypothetical complex structure as a classical ground state of a yangmillshiggslike theory on s6. Complex manifolds without potential theory chern s.
Chern, complex manifolds without potential theory springer ver lag, berlin. Math 545 topology and geometry of manifolds winter 2000. Given a complex manifold x, we can think about xwithout its complex structure. Namely, they make either the complex numbers, in which case n 2 and there is a single complex structure j, or the quaternions, in which case n 2 and thre are three possible imaginary units j1,j2,j3. Recently chern proposed 5 that noneof the almost complex structures. However, many mathematicians never take more than an. Harmonic forms and hodge theory on riemannian manifolds. Pdf we introduce the notion of a special complex manifold. All of this structure is reflected in a rich theory of geometric and topological invariants. Narasimhan no part of this book may be reproduced in any form by print, micro. The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and. Complex manifolds without potential theory springerlink. Morozov and perelomov, string theory and complex geometry, phys.
Complex manifolds without potential theory eells 1980. Chern, complex manifolds without potential theory 4. Subharmonic functions are an essential tool of harmonic analysis and potential theory. In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in c n, such that the transition maps are holomorphic the term complex manifold is variously used to mean a complex manifold in the sense above which can be specified as an integrable complex manifold, and an almost complex manifold. Complex analytic and differential geometry institut fourier. There are surprisingly rich properties of these holomorphic functions. I certify that this is an original project report resulting from the work completed during this period. There is no question that complex analysis is a corner stone of the analysis education at every university and each area of mathematics requires at least some knowledge of it. This classical vacuum solution is then constructed by fourier expansion dimensional reduction from the obvious one of a similar theory. This volume serves as an introduction to the kodairaspencer theory of deformations of complex structures. Complex manifolds without potential theory with an. Yangmills, complex structures and cherns last theorem.
A classical invitation to algebraic numbers and class fields curtis. The deformation theory of almost complex structures is described by the kodairaspencer theory which we note, is also relevant to the topological typeb string theory 10. With an appendix on the geometry of characteristic classes universitext on. On geometric quantization of compact, complex manifolds. Potential theory on almost complex manifolds numdam. With an appendix on the geometry of characteristic classes, second edition universitext. Math 549 geometric structures spring 2004 suggested reading books about complex manifolds with the exception of m, which can be downloaded from the internet, these books are or soon should be on reserve in the math research library. Simple to complex with some numerical computations, was completed by mr. Theorems 1 and 4 below extend without any changes to quasiregular maps of surfaces. Complex manifolds lecture notes based on the course by lambertus van geemen a. You can read online complex manifolds without potential theory here in pdf, epub, mobi or docx formats.
The purpose of this paper is to develop an intrinsic potential theory. Math 545 topology and geometry of manifolds winter 2000 suggestions for further reading. Ii tseng, lisheng and yau, shingtung, journal of differential geometry, 2012. Potential theory on almost complex manifolds department.
Complex manifolds and hermitian differential geometry. Cohomology and hodge theory on symplectic manifolds. Demailly, complex analytic and differential geometry. Chern, complex manifolds without potential theory, second edition, springerverlag, 1979. Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s.
Geometric pluripotential theory on k ahler manifolds. Pseudoholomorphic curves on almost complex manifolds have been much more intensely studied than their dual objects, the plurisubharmonic functions. The ones marked may be different from the article in the profile. Its brief history and its future personal perspective shingtung yau once complex number is introduced as a. Potential theory on almost complex manifolds article pdf available in annales institut fourier 651 july 2011 with 47 reads how we measure reads. Complex structure on the six dimensional sphere from a. The differential geometrical methods of this theory were developed essentially under the influence of professor s. From holomorphic functions to complex manifolds graduate. Chern, complex manifolds without potential theory, new york, springerverlag, 1979.
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