The hodge conjecture belongs to topology and geometry. One objective is to present, in a general context, some of the recent work of carayol c1, c2, c3. Hodge theory and complex algebraic geometry ii by claire. Download freecourseweb complex algebraic geometry torrent. Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Hodge theory and complex algebraic geometry ii by claire voisin. Hodge decomposition for curves let x be a compact connected complex curve. According to the beautiful conjectures of hodge, bloch and beilinson. X the relative cohomology groups hkx, y have a mixed hodge structure and the long exact cohomology sequence. In general this group is di cult to understand, but when xis k ahler we can use hodge theory to show that it can be elegantly described as a disjoint union of complex tori. Current developments in algebraic geometry msri publications volume 59, 2011 the hodge theory of character varieties mark andrea a. The method rests on new hodge theoretic results on the cohomology of projective varieties which extend naturally the classical. The meanings of these results are investigated in several directions.
The book is is completely selfcontained and can be used by students, while its content gives an uptodate account of hodge theory and complex algebraic geometry. Algebraic geometry over the complex numbers springerlink. This is a modern introduction to kaehlerian geometry and hodge structure. It can be used as a main text for a second semester graduate course in algebraic geometry with emphasis on sheaf theoretical methods or a more advanced graduate course on algebraic geometry and hodge theory. Cambridge core algebra hodge theory and complex algebraic geometry i by claire voisin. Complex manifolds and kahler geometry prof joyce 16 mt. The algebraic techniques are currently the dominant ones. The book culminates with the hodge decomposition theorem.
Index 349 deligne invariant cycles theorem, 124 deligne mixed hodge structure on the relative cohomology, 220 divisor cartier, 35, 90, 248, 251, 252 exceptional, 271 modulo rational equivalence, 228. Cambridge core geometry and topology hodge theory and complex algebraic geometry ii by claire voisin. Download pdf algebraic geometry i ebook full get book. This proof gives the result for rational cohomology under the hypothesis that y and x are smooth. Hodge theory and complex algebraic geometry 1 claire. Hodge theory, complex geometry, and representation theory. Download pdf algebraic geometry i ebook full get book direct. Voisins books hodge theory and complex algebraic geometry, i.
Hodge theory and complex algebraic geometry i, claire voisin, isbn. As is well known, one the main conjectures guiding this research is the generalized hodge conjecture on the realization of every rational hodge cohomology class as an. This manuscript is based on lectures given by steve shatz for the course math 622623complex algebraic geometry, during fall 2003 and spring 2004. A unifying theme is the geometry of homogeneous complex manifolds. Hodge theory and complex algebraic geometry i volume 1 cambridge studies in advanced mathematics. The material presented here consists of a more or less selfcontained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or riemann surfaces. As applications, we give a closed explicit formula for certain canonical. The second part is concerned with the definitions and basic properties of projective space in n dimensions. Kodaira, complex manifolds and deformation of complex structures, springer, 1986. Higgs bundles, integrability, and holomorphic forms dvi file published in motives, polylogarithms and hodge theory, intl press 2002. Hodge theory is very powerful in complex geometry, and my impression from attending talks is that padic hodge theory is really number theorists hoping to have the same machinery in the arithmetic setting. Cambridge studies in advanced mathematics includes bibliographical references and index. The slides are for a lecture given at the workshop on algebraic geometry and geometric modeling to be. It also gives a framework in which to describe energyconserving flows, and so.
The relationship between hodge theory and geometry, especially for the study of algebraic cycles on complex manifolds, is a rich, deep and increasingly active field. Hodge theory and complex algebraic geometry 2 pdf free. Hodges primary motivation, the study of complex projective varieties, is encompassed by the latter case. Algebraic geometry over the complex numbers is intended for graduate level courses in algebraic geometry and related fields. Hodge s primary motivation, the study of complex projective varieties, is encompassed by the latter case. Voisins books hodge theory and complex algebraic geometry, i, ii, therefore, the reader will not. Finite dimensional representation theory interacts with hodge theory through the use of hodge representations to classify the possible realizations of a reductive, q algebraic. Hodge theory problem sheet 1 these exercises were pulled from the following books.
Hodge theory and geometry this expository paper is an. Voisin, hodge theory and complex algebraic geometry i. The last part deals with the relationships between hodge theory and algebraic cycles. Hodge theory and complex algebraic geometry i by claire voisin. Volume 1 cambridge studies in advanced mathematics by claire voisin hodge theory and complex algebraic geometry i. Entire holomorphic mappings in one and several complex variables. The first of two volumes offering a modern introduction to kaehlerian geometry and hodge structure.
Voisin, hodge theory and complex algebraic geometry, i, cambridge studies in advanced mathematics 76, cambridge university press, 2002, chapters. The text is complemented by exercises giving useful results in complex algebraic geometry. Entdecken sie hodge theory and complex algebraic geometry i. This course will present the basics of hodge theory aiming to familiarize students with an important technique in complex and algebraic geometry. The 2003 second volume of this selfcontained account of kaehlerian geometry and hodge theory continues voisins study of topology of families of algebraic varieties and the relationships between read more. Download pdf hodge theory and complex algebraic geometry. Additional preprintstalks some applications of positive characteristic techniques to vanishing theorems pdf file, talk at lipman conf.
In its simplest form, hodge theory is the study of periods integrals of algebraic di erential forms which arise in the study of complex geometry, number theory and physics. Today, algebraic geometry is applied to a diverse array of. Jul 02, 1999 the second volume of this modern account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. Algebraic geometry played a central role in 19th century math.
For a more serious introduction, you can get my notes on basic algebraic geometry. I algebraic birational geometry, singularity theory, geometric invariant theory git, etc. We give a geometric proof of the decomposition theorem of beilinson, bernstein, deligne and gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. I will require students to take turns signing up to record, in texed format, what i said in. Phillip augustus griffiths iv is an american mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. We aim to present materials which are not covered in j. The text is complemented by exercises which provide useful results in complex algebraic geometry. Hodge decomposition let us return brie y to the case of x a smooth projective algebraic curve of genus g. Its di culty and richness arises in part from the nonalgebraicity of these integrals. Lewiss book a survey of the hodge conjecture and c.
Martin luther preacher of the cross a guide to computer user support for help desk and support specialists eyes on russia developmental mathematics 7th edition. The turn of the 20th century saw a sharp change in attitude to algebraic geometry. Hodge theory and complex algebraic geometry claire voisin. In between, the author proves the kaehler this is a modern introduction to kaehlerian geometry and hodge structure.
Introduction to hodge theory daniel matei snsb 2008 abstract. For applications to algebraic geometry one needs the harmonic theory to play well with the complex geometry e. He was a major developer in particular of the theory of variation of hodge structure in. Migliorini, where we prove, for each of the groups gl c. The book concludes with the example of cycles on abelian varieties, where some results of bloch and beauville, for example, are expounded.
Topics covered include the quintic threefold, toric geometry, hodge theory, complex and kahler moduli, gromovwitten invariants, quantum cohomology, localization in equivariant cohomology, and the work of lianliuyau and givental on the mirror theorem. The process for producing this manuscript was the following. Migliorini, where we prove, for each of the groups gl. First, by using operators from hodge theory on compact kahler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. This course will talk about the elementary theory in this subject such as complex manifolds, kahler geometry, projective varieties, sheaf theory and hodge decomposition theorem. The main results are the generalized noetherlefschetz theorems, the generic triviality of the abeljacobi maps, and most importantly, noris connectivity theorem, which generalizes the above. Math 448 or permission of the instructor this course provides an introduction to complex manifolds, especially compact kahler manifolds. Hodge theory and complex algebraic geometry find, read and cite all the. Its a 340k pdf file my notes algebraic geometry over the complex numbers covers more. The approach adopted in this course makes plain the similarities between these different.
Also included is an introduction to hodge theory and intersection homology based on the simple topological ideas of lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory. The set of all holomorphic line bundles on a complex manifold xforms an interesting group picx. It starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory and culminates with the hodge decomposition theorem. Hodge theory and complex algebraic geometry, i cambridge studies in advanced mathematics 76 burt totaro. Hodge theory and complex algebraic geometry ii cambridge studies in advanced mathematics v. Volume 1 cambridge studies in advanced mathematics by claire voisin pdf, epub ebook d0wnl0ad this is a modern introduction to kaehlerian geometry and hodge structure. The last part of the book is devoted to the relationships between hodge theory and algebraic cycles. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more. Semantic scholar extracted view of hodge theory and complex algebraic geometry ii by claire voisin et al. The deepest results of abel, riemann, weierstrass, and many of the most important works of klein and poincare were part of this subject. H huybrechts, complex geometry v voisin, hodge theory and complex algebraic geometry, i you might nd more information if you consult the book itself. Volume 2 by claire voisin, 9780521718028, available at book depository with free delivery worldwide. We want to understand the basic idea behind the proof of theorem 1. Algebraic geometry verses complex algebraic geometry.
The text is complemented by exercises offering useful results in complex algebraic geometry. We treat two quite different problems related to changes of complex structures on kahler manifolds by using global geometric method. Note that these notes will be posted publicly online, in both tex and pdf. The 2003 second volume of this account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. This volume contains the proceedings of an nsfconference board of the mathematical sciences cbms regional conference on hodge theory, complex geometry, and representation theory, held on june 18, 2012, at the texas christian university in fort worth, tx. Courses like introductory topology, differential geometry, algebraic topology, riemannian geometry, complex manifolds, and differential topology will be critical to understanding the mathematics necessary for hodge theory and the hodge conjecture. Algebraic geometry and hodge theory 2017spring letters of recommendation policy this course is an introduction to complex algebraic geometry and hodge theory, which is for mathematicians intending to get near the boundary of current research, in complex geometry or a related part of mathematics. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros the fundamental objects of study in algebraic geometry are algebraic varieties, which are. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory with the latter being treated in a more theoretical way than. Computer scientists have made extensive use combinatorics, graph theory, probability, and linear algebra. For this, we need a kahler metric as described in alex degtyarevs answer. The second volume of this modern account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. Completely selfcontained, the book is ideal for students, while its content gives an account of hodge theory and complex algebraic geometry as has been developed by p.
Here are some postscript or pdf files containing lecture notes for various lectures given between 2001 and 2012. A course in hodge theory bienio da matematica impa. Hodge theory and complex algebraic geometry ii semantic scholar. Formal hodge theory the basic objects of formal hodge theory discussed. The author proves the kaehler identities, which leads to the hard lefschetz theorem and the hodge. 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. The class of k ahler manifolds include all smooth projective varieties over the complex numbers but is more general, including. It takes place in even dimensions, and can be considered a version of complex algebraic.
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