Hodge Theory (eBook)

Eduardo Cattani, Phillip A. Griffiths, LE DUNG TRANG, Fouad El Zein (Autoren)

eBook Download: PDF | EPUB

2014
608 Seiten
Princeton University Press (Verlag)
978-1-4008-5147-8 (ISBN)

Infos

Eduardo Cattani is professor of mathematics at the University of Massachusetts, Amherst. Fouad El Zein is a researcher at the Institut de Mathématiques de Jussieu, Université de Paris VII. Phillip A. Griffiths is former director and professor emeritus of mathematics at the Institute for Advanced Study in Princeton. Lê D?ng Tráng is professor emeritus of mathematics at the Aix-Marseille Université.

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This book provides a comprehensive and up-to-date introduction to Hodge theory-one of the central and most vibrant areas of contemporary mathematics-from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research.The book is divided between introductory and advanced lectures. The introductory lectures address Kahler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.The contributors include Patrick Brosnan, James Carlson, Eduardo Cattani, Francois Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark L. Green, Phillip A. Griffiths, Matt Kerr, Le Dung Trang, Luca Migliorini, Jacob P. Murre, Christian Schnell, and Loring W. Tu.

Autor

Eduardo Cattani is professor of mathematics at the University of Massachusetts, Amherst. Fouad El Zein is a researcher at the Institut de Mathématiques de Jussieu, Université de Paris VII. Phillip A. Griffiths is former director and professor emeritus of mathematics at the Institute for Advanced Study in Princeton. Lê Dũng Tráng is professor emeritus of mathematics at the Aix-Marseille Université.

Produkt-Details

• Erscheint lt. Verlag: 21.7.2014
• Reihe/Serie: Mathematical Notes
• Reihe/Serie: Mathematical Notes
• Zusatzinfo: 10 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• Schlagworte: Abelian category • abelian group • abelian variety • Abel–Jacobi map • AbelЊacobi map • absolute Hodge classes • abstract variations • Addition • Adélic lemmas • Adlic lemmas • Albanese kernel • Alexander Grothendieck • Algebraic Curve • algebraic cycle • Algebraic cycles • algebraic equivalence • Algebraic Geometry • algebraic homology • algebraicity • algebraic maps • Algebraic Structure • Algebraic Varieties • algebraic variety • Analytic manifold • asymptotic behavior • automorphism • Base change • bilinear form • Bloch–Beilinson conjecture • BlochЂeilinson conjecture • Čech cohomology • Chow group • Chow groups • Codimension • coefficient • coherent sheaves • cohomology • Cokernel • Compactification (mathematics) • compact Kähler manifolds • compact Khler manifolds • Complex Analysis • Complex dimension • complex manifold • complex manifolds • complex multiplication • complex number • Complex torus • conjectural filtration • conjecture • contemporary mathematics • Covering space • Cup product • cycle class • Cycle map • decomposition theorem • Deligne cohomology • Deligne's theorem • de Rham cohomology • de Rham theorem • derived category • Diagram (category theory) • differentiable manifold • differential form • differential forms • Dimension (vector space) • direct sum • Divisor • _ech cohomology • Elliptic Curve • Elliptic Curves • Embedding • Endomorphism • equivalence relation • equivalence relations • exact sequence • existential quantification • Exterior algebra • fiber bundle • functor • Fundamental class • fundamental group • Galois action • Geometry • Grassmannian • Griffiths group • Griffiths' period map • Grothendieck's theorem • harmonic forms • Hermitian structures • Hermitian symmetric domains • Hodge bundles • Hodge conjecture • Hodge cycles • Hodge structure • Hodge structures • Hodge-theoretic interpretations • Hodge Theory • holomorphic function • holomorphicity • homological equivalence • Homomorphism • Homotopy • horizontal distribution • horizontality • hypercohomology • hyperplane • hypersurface • hypersurfaces • intersection cohomology complex • intersection cohomology groups • Intersection form (4-manifold) • invariant cycle theorem • Invertible matrix • Irreducible component • isomorphism class • Jacobian ideal • Kähler manifolds • Kähler structures • Khler manifolds • Khler structures • Kodaira–Spencer map • KodairaГpencer map • Kuga–Satake correspondence • KugaГatake correspondence • Lefschetz decomposition • Leray spectral sequence • Lie algebra • linear algebra • Linear map • line bundle • Local System • Local Systems • Mathematical Induction • mixed Hodge complex • Mixed Hodge structure • mixed Hodge structures • Module (mathematics) • Monodromy • Morphism • Morphisms • Natural number • Nilpotent orbit • nontrivial topological constraints • Normal function • normal functions • Open set • period domains • period mappings • perverse sheaf • Poincaré residues • Poincar residues • projective space • projective variety • Quasi-projective variety • Schmid's orbit theorems • sheaf cohomology • Sheaf (mathematics) • shimura varieties • smooth case • Smoothness • smooth projective varieties • Special case • Spectral Sequence • spectral sequences • spread philosophy • Spreads • SUBGROUP • submanifold • Subset • Summation • Surjective function • Symplectic Structures • Tangent Space • Tangent vector • tensor product • Theorem • Thom–Whitney results • ThomЗhitney results • topological invariants • Topological space • Topology • Torelli theorem • Transversality (mathematics) • variational Hodge conjecture • vector bundle • Vector field • Vector Space • Verdier duality • Zariski topology
• ISBN-10: 1-4008-5147-5 / 1400851475
• ISBN-13: 978-1-4008-5147-8 / 9781400851478