Hodge theory is a powerful tool in analytic and algebraic 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. The three main topics are: $L^2$ Hodge theory and vanishing theorems; Hodge theory in characteristic $p$; and variations of Hodge structures and mirror symmetry. Each section has a detailed introduction and numerous references. Many open problems are also included. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.
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.
This is a version of the Jacobson— Morosov theorem. EXERCISE A.3.7 Let N C g[(V) be nilpotent and Y C g[(V) be semisimple. Set V4 I E4 Then the following are equivalent: (1) There exists an slg—triple {N+, NI, Y} with N+ I N. (2) The ...
This is a modern introduction to Kaehlerian geometry and Hodge structure.
Offers an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann; and then the studies of two-dimensional multiple integrals by Poincare and Picard.
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The main goal of the CIME Summer School on "Algebraic Cycles and Hodge Theory" has been to gather the most active mathematicians in this area to make the point on the present state of the art.
The text is complemented by exercises which provide useful results in complex algebraic geometry.
James Carlson, Stefan Müller-Stach, Chris Peters. Lu, X. and K. Zuo. 2014. On Shimura curves in the Torelli locus of curves. arxiv.org/abs/1311.5858. Lu, X. and K. Zuo. 2015. The Oort conjecture on Shimura curves in the Torelli locus of ...
Combines cutting-edge research and expository articles in Hodge theory. An essential reference for graduate students and researchers.
Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)