Hodge theory and complex algebraic geometry 1 claire. The aim of this note is to clarify a minor mistake in the article. Hodge decomposition university of california, berkeley. In particular, in general relativity the term pseudotensor is sometimes used, where neither of the above meanings are implied. In order to define the so called hodgede rham operator one needs the adjoint of this differential. The symplectic geometry part of the course follows the book by ana cannas da silva, lectures on symplectic geometry lecture notes in mathematics 1764, springerverlag. A variant of the hodge theorem is the hodge decomposition. The hodge index theorem, hodge structures and polarizations are discussed. One then deduces both results by reducing to the analytic case we did this for hodge decomposition above.
This thesis describes the hodge decomposition of the space of differential forms on a. It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry. Numerous and frequentlyupdated resource results are available from this search. He develops the theory without restricting to orientable manifolds. Pdf the hodgede rham theorem is introduced and discussed. These relations allow one to think of the hodge star \\ as an operator that roughly swaps the exterior and inner products, or alternatively that yields the orthogonal complement with the same magnitude. Complex analytic and differential geometry, opencontent book. However, to show that this indeed gives an example of a hodge complex follows only after a strong form of the hodge decomposition is shown to hold. This of course depends on the choice of metric, while the cohomology groups are. Hodge theory, elliptic differential operators and jacobis. We then introduce laplacians and establish the connection between harmonic forms and cohomology.
Lectures on the hodgede rham theory of the fundamental group of p1 f0. Nygaard used the drw theory to give an easier proof of the rudakovshafarevich vanishing theorem on k3surfaces, and to generalize ogusmazurs theorem on. In mathematics, the hodgede rham spectral sequence named after w. Introduction to hodge theory 3 the decomposition 1.
A hodge structure is said to be of weight d d if it. Partial collapsing and the spectrum of the hodgede rham. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory with the latter being treated in a more theoretical way than is usual in geometry. 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. Hodge theory harvard mathematics harvard university. In particular, it gives an isomorphism referred to as the hodge decomposition. Wells and the explanation of gil cavalcanti about the subjects covered. Dec 06, 2014 now, heres the hard part of the proof. This allows us to deduce information about forms from topological properties. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. Request pdf the fischer decomposition for hodgede rham systems in euclidean spaces the classical fischer decomposition of spinorvalued polynomials is a. Let m1 and m2 be two connected compact manifolds with the same boundary. This book is a writtenup and expanded version of eight lectures on the hodge theory of projective manifolds.
January 1998, volume 1, number 622 second of 4 numbers. We strongly urge the reader to read this online at instead of reading the old material. Im reading the proof of the hodge decomposition theorem in warner, foundations of differentiable manifolds and lie groups. 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. This result has implications for the general study of several partial differential. Part of the graduate texts in mathematics book series gtm, volume 94. In section 2, we discuss the spectral theory of selfadjoint elliptic partial differential operators and give the hodge decomposition theorem. Hodge decomposition for manifolds with boundary and. A course in hodge theory bienio da matematica impa. However, to show that this indeed gives an example of a hodge complex follows only after a strong form of the hodge decomposition is. Hodge theory is a powerful tool in analytic and algebraic geometry. Now we can go about proving the helmholtz decomposition.
Our main objective in this chapter is a proof of the hodge decomposition theorem. This is a modern introduction to kaehlerian geometry and hodge structure. This brief discussion assumes some basic knowledge of the hodgede rham theory, and can be skipped by readers only. Limit mixed hodge structures and the drinfeld associator 37 references 44 date.
Among the goals of this paper are to give a context to manins iterated shimura integrals iterated integrals of modular forms and to study. I suppose we have to ask the riemannian manifold to be complete. Richard hain submitted on 25 mar 2014, last revised 11 jul 2015 this version, v3. Introduction to hodge theory jose bertin, jeanpierre. The use of pseudo to indicate a quantity that reverses sign upon a change of orientation should not be confused with the use of pseudo to indicate an inner product that is not positivedefinite. For any riemannian manifold, there is a theorem also called hodge decomposition. Introduction to hodge theory download pdfepub ebook.
As in the wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic di erential forms. One then deduces both results by reducing to the analytic case. The book culminates with the hodge decomposition theorem. Harmonic differential forms and the hodge decomposition theorem. In section 3, we define the classical elliptic complexes. The main objective of the present book is to give an introduction to hodge theory and its main conjecture, the socalled hodge conjecture. The meanings of these results are investigated in several directions. This book features an approach that employs recursive arguments on dimension and does not introduce spaces of higher. The most we can say is that the hodge decomposition provides us with a privileged representative of the cohomology class, namely, the harmonic form. The fischer decomposition for hodgede rham systems in euclidean spaces article in mathematical methods in the applied sciences 351 january 2012 with 15 reads how we measure reads. 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. If the address matches an existing account you will receive an email with instructions to reset your password. For completeness, we discuss the hodge decomposition in section 3. Some applications of the hodgede rham decomposition to ricci solitons.
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