Hodge theory and complex algebraic geometry 1 claire voisin, leila schneps. Hodge theory and complex algebraic geometry find, read and cite all the. Voisins books hodge theory and complex algebraic geometry, i. Hodge theory and complex algebraic geometry 1 pdf free. This is a modern introduction to kaehlerian geometry and hodge structure. The text comes in two parts that correspond to the distribution. Algebraic geometry over the complex numbers springerlink. 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. Download complex geometry ebook in pdf, epub, mobi. I dont think the hodge numbers are topological invariants. Lewiss book a survey of the hodge conjecture and c. A course in hodge theory bienio da matematica impa. The text is complemented by exercises which provide useful results in complex algebraic geometry. Proofs of the lefschetz theorem on hyperplane sections, the picardlefschetz study of lefschetz pencils, and deligne theorems on the degeneration of the leray spectral sequence and the global invariant cycles follow.
We want to understand the basic idea behind the proof of theorem 1. 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. Hodge theory and complex algebraic geometry find, read and cite all the research you need on researchgate. For applications to algebraic geometry one needs the harmonic theory to play well with the complex geometry e. It covers sheaf theory, cohomology, some hodge theory, as well as some of the more algebraic aspects of algebraic geometry. An introduction to hodge theory and applications to algebraic geometry lynnelle ye june 3, 2015 1 introduction in this article, we develop the basic ideas behind the hodge theorem on harmonic forms and the hodge identities and decomposition for k ahler manifolds. The first of two volumes offering a modern introduction to kaehlerian geometry and hodge structure. Hodge theory and complex algebraic geometry 1 and 2. It starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory and culminates with the hodge decomposition theorem. Cambridge core algebra hodge theory and complex algebraic geometry i by claire voisin.
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. The last part of the book is devoted to the relationships between hodge theory and algebraic cycles. Introduction to hodge theory 3 the decomposition 1. Buy hodge theory and complex algebraic geometry, i. Hodge theory and complex algebraic geometry i claire voisin. For example, it enables us at least partially to study moduli spaces classifying the deformations of the complex structure on a polarised algebraic variety, and possibly, when the period map is injective, to realise these. Voisins books hodge theory and complex algebraic geometry, i, ii, therefore, the reader will not. 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. Hodge decomposition let us return brie y to the case of x a smooth projective algebraic curve of genus g. The readable download signals how those studies are written enough and notified. Though the proof of the hodge theorem is omitted, its consequences topological, geometrical and algebraic are discussed at some length.
Cohomology of coherent sheaves on complex algebraic. Algebraic geometry over the complex numbers is intended for graduate level courses in algebraic geometry and related fields. Hodge theory and complex algebraic geometry ii byvoisin. 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. Download hodge theory and complex algebraic geometry. On certain boundedness of fibred calabiyau s threefolds. According to a conjecture of deligne and grothendieck, there is a unique natural transformation csm from the functor of constructible functions on complex algebraic varieties to the homology of complex. For example, it enables us at least partially to study moduli spaces classifying the deformations of the complex structure on a polarised algebraic variety, and possibly, when the period map is injective, to realise these moduli spaces as subspaces of domains of global. Volume 1 cambridge studies in advanced mathematics book 76 kindle edition by voisin, claire, schneps, leila. Hodge theory and complex algebraic geometry ii by claire voisin.
Pdf hodge theory and algebraic geometry semantic scholar. The book culminates with the hodge decomposition theorem. A unifying theme is the geometry of homogeneous complex manifolds. Algebraic geometry over the complex numbers donu arapura. Note hodge theory and algebraic geometry 200210711 department of mathematics, hokkaido university nash problem on arc families for singularities. The 2003 second volume of this account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. 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 author then proves the kaehler identities, which leads to the hard lefschetz.
Cambridge core geometry and topology hodge theory and complex algebraic geometry ii by claire voisin. This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. Hodge theory and complex algebraic geometry i, ii, by claire voisin. The method rests on new hodgetheoretic results on the cohomology of projective varieties which extend naturally the. In between, the author proves the kaehler this is a modern introduction to kaehlerian geometry and hodge structure. An introduction to hodge theory and applications to. Proofs of the lefschetz theorem on hyperplane sections, the picardlefschetz study of lefschetz pencils, and deligne theorems on the degeneration of the. Ii, therefore, the reader will not find in this book some of the. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory with the latter being treated in a more theoretical way than. Are there free legal equivalent books or freely available lecture notes or courses of claire voisin. Finite dimensional representation theory interacts with hodge theory through the use of hodge representations to classify the possible realizations of a reductive, qalgebraic. Hodge theory and complex algebraic geometry claire.
Hodge theory and complex algebraic geometry 1 claire. This page intentionally left blank cambridge studies in advanced mathematics 76 editorial board. Deligne and gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. Note that these notes will be posted publicly online, in both tex and pdf form. Read hodge theory and complex algebraic geometry ii byvoisin by voisin for online ebook.
Kang the smooth center of the cohomology of a singular variety published in hodge theory, complex geometry, and representation theory, ams 2014 decomposition of direct images of logarithmic differentials internat. We compute the behaviour of hodge data by tensor product with a unitary rankone local system and middle convolution by a kummer unitary rankone local system for an irreducible variation of polarized complex hodge structure on a punctured complex affine line. We then use these ideas to obtain results such as the kodairanakano vanishing. Hodge theory and complex algebraic geometry ii cambridge studies in advanced mathematics 77 burt totaro. Hodge theory and complex algebraic geometry article pdf available in proceedings of the edinburgh mathematical society 4901 february 2006 with 1,027 reads how we measure reads. Hodge theory and complex algebraic geometry ii by claire. 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.
The book concludes with the example of cycles on abelian varieties, where some results of bloch and beauville, for example, are expounded. 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. Hodge theory of cyclic covers branched over a union of hyperplanes can j. Use features like bookmarks, note taking and highlighting while reading hodge theory and complex algebraic geometry i.
Volume 1 cambridge studies in advanced mathematics by claire voisin hodge theory and complex algebraic geometry i. Hodge theory and complex algebraic geometry i by jayna. This course will talk about the elementary theory in this subject such as complex manifolds, kahler geometry, projective varieties, sheaf. Voisin, hodge theory and complex algebraic geometry, i, p. The author then proves the kaehler identities, which leads to. 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.
Hodge theory and complex algebraic geometry ii cambridge. Hodge theory and complex algebraic geometry i by claire voisin. This proof gives the result for rational cohomology under the hypothesis that y and x are smooth. Cambridge studies in advanced mathematics includes bibliographical references and index.
Consider the category of complex algebraic varieties with proper morphisms. We aim to present materials which are not covered in j. Hodge theory and complex algebraic geometry, i cambridge studies in advanced mathematics 76 burt totaro. Download it once and read it on your kindle device, pc, phones or tablets. The second volume of this modern account of kaehlerian geometry and hodge theory starts with the topology of families of algebraic varieties. Hodge theory and complex algebraic geometry claire voisin. Everyday low prices and free delivery on eligible orders.
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