A course in hodge theory bienio da matematica impa. Hodge decomposition university of california, berkeley. The text comes in two parts that correspond to the distribution. For the first time ever, this book collects together all the results obtained in the last few years which aim to characterise those infinite groups which can.
For completeness, we discuss the hodge decomposition in section 3. On the smooth deformation of hilbert space decompositions j. Elworthy and xuemei li for a compact riemannian manifold the space l2a of l2 di. Part of the graduate texts in mathematics book series gtm, volume 94. Pdf stability of spectra of hodgede rham laplacians. Hodge structure academic dictionaries and encyclopedias. The hodge decomposition theorem, which lies at the heart of hodge theory, uses these operators to decompose the space of di erential kforms into a direct sum of 5. All of the content in these notes in contained in the book differential analysis. If m is a closed, oriented riemannian nmanifold, the hodge decomposition theorem. Our main objective in this chapter is a proof of the hodge decomposition theorem.
In every aspect, this is an excellent book, invaluable both for learning the topic and a reference. Hodge theory, elliptic differential operators and jacobis. It is a cohomology theory based on the existence of differential forms with. Among the goals of this paper are to give a context to manins iterated shimura integrals iterated integrals of modular forms and to study relations in the. This book consists of expositions of aspects of modern hodge theory, with the purpose of providing the nonexpert reader with a clear idea of the current state of the subject. We establish unisolvence and a geometric decomposition of the space. On a given complex manifold x, there are two natural cohomologies to consider. Given a scheme in characteristic p together with a lifting modulo p2, we construct a functor from a category of suitably nilpotent modules with connection to. Kleine ag degeneration of the hodgetode rham spectral. Download for offline reading, highlight, bookmark or take notes while you read hodge theory mn49. The hodgede rham theorem is introduced and discussed. A variation of hodge structure is a family of hodge structures parameterized by a manifold, first studied by phillip griffiths 1968.
Among the goals of this paper are to give a context to manins iterated shimura integrals iterated integrals of. In section 3, we define the classical elliptic complexes. A variant of the hodge theorem is the hodge decomposition. The main objective of the present book is to give an introduction to hodge theory and its main conjecture, the socalled hodge conjecture. Three lectures on hodge structures damien m egy june 2, 2014 these are preliminary notes for three lectures given at the spring school \classical and p. This brief discussion assumes some basic knowledge of the hodgede rham theory, and can be skipped by readers only. Thus a context in which the hodge star is often seen is the exterior algebra of the cotangent bundle, the space of differential forms on a manifold, where it can be used to construct the codifferential from the exterior derivative, and thus the laplacede rham operator, which leads to the hodge decomposition of differential forms in the case. We strongly urge the reader to read this online at instead of reading the old material. Introduction to hodge theory 3 the decomposition 1. Proving hodge decomposition without using the theory of. The book by ljusternik and sobolev 1961 contains a moreuptodate treatment. Some results on halmost ricci solitons sciencedirect. The hodge decomposition theorem on compact manifold is also studied in the context of conserved charges. Harmonic differential forms and the hodge decomposition.
The use of the sobolev topology strikingly alters the problem from the classical setup and gives rise to a new class of elliptic boundary value problems. On the other hand, given a vector field x over a compact oriented riemannian manifold m n the hodgede rham decomposition theorem, see e. This book is an exposition of what is currently known about the fundamental groups of compact kahler manifolds. This books is an extensive and comprehensive survey on pseudoriemannian submanifolds and. This allows us to deduce information about forms from topological properties. In the common hodge theory books, the authors usually cite other sources for the theory of elliptic operators, like in the book about hodge theory of claire voisin, where you find on page 128 the theorem 5. In his thesis goldring 1977 has completed the first steps of a hodge decomposition theorem for nonlinear operators when regarded as differential one forms. E f be an elliptic differential operator on a compact manifold. This result has implications for the general study of several partial.
I suppose we have to ask the riemannian manifold to be complete. A crucial point is that the hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral or rational coefficients. Complex analytic and differential geometry, opencontent book. Hodge decomposition an overview sciencedirect topics. Nonabelian hodge theory in characteristic p internet archive. I wanted to briefly address the question of how much of the hodge decomposition can be understood algebraically. In order to define the so called hodgede rham operator one needs the adjoint of this differential.
The second objective is to give a reminder on the basic of deformation theory of smooth schemes. Philosophically, a motif is the cohomology essence of a variety. Heat equation, jacobis elliptic and theta functions. Pdf the hodgede rham theorem is introduced and discussed. The sections of this bundle are differential forms, to be noted by. Limit mixed hodge structures and the drinfeld associator 37 references 44 date. They appear in the thick of books where at last one of those subjects is considered. Hodge dual academic dictionaries and encyclopedias. The original papers of deligne often provide a more conceptual formalism which applies in both the complex and arithmetic settings. Stability of spectra of hodgede rham laplacians 337 lemma 4. The cuspid of his results is constituted by the following. This book is a study of the hodgekodaira and related. Hodge decomposition for manifolds with boundary and. This class of groups contains all finite groups and is strictly smaller than the class of all finitely presentable groups.
Kahler geometry and hodge theory olivier biquard and andreas horing the aim of these lecture notes is to give an introduction to analytic geometry, that is the geometry of complex manifolds, with a focus on the hodge theory of compact kahler manifolds. In mathematics, the hodge star operator or hodge dual is a significant linear map introduced in general by w. This author has made progress in relating those two theorems. From it information on the geometry and sometimes the topology of the underlying spaces can be obtained. Hodge theory is a powerful tool in analytic and algebraic geometry. Hodge decomposition for manifolds with boundary and vector. Aspects of differential geometry ii synthesis lectures on. Hhd, classifying some important books and articles into. Hodge decomposition a method for solving boundary value problems lecture notes in mathematics hodge theory is a standard tool in characterizing differ ential complexes and the topology of manifolds. The model for selfdual chiral bosons as a hodge theory. Contents preface ix 1 introduction 1 2 basic notions of homological algebra 11 2.
This result has implications for the general study of several partial differential. It is defined on the exterior algebra of a finite dimensional oriented inner product space. Rhams cohomology of soft hypersurfaces griffithss theorem. Gunter schwarz hodge theory is a standard tool in characterizing differ ential complexes and the topology of manifolds. As in the wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic di erential forms. We will use it to give a proof of the poincar e duality in a compact oriented manifold. Hodge theory mn49 ebook by eduardo cattani rakuten kobo. The classical theorem of derham 1931 asserts that the. The hodge decomposition theorem, which lies at the heart of hodge theory, uses these operators to. Lectures on the hodgede rham theory of the fundamental group of p1 f0. A mixed hodge structure is a generalization, defined by pierre deligne 1970, that applies to all complex varieties even if they are singular and noncomplete.
234 385 551 79 756 816 22 1516 445 297 1235 108 803 1607 1026 1133 1482 409 929 1209 631 1295 140 689 494 1210 1390 335 937 677 825 1224 41 1174 119 1602 863 1298 1489 746 406 1056 362 138 428 389