Hodge concept originated as an software of harmonic thought to the research of the geometry of compact advanced manifolds. the information have proved to be fairly robust, resulting in essentially very important effects all through algebraic geometry. This publication includes expositions of assorted points of contemporary Hodge idea. Its objective is to supply the nonexpert reader with an exact notion of the present prestige of the topic. the 3 chapters improve precise yet heavily similar matters: $L^2$ Hodge idea and vanishing theorems; Frobenius and Hodge degeneration; diversifications of Hodge buildings and replicate symmetry.The strategies hired disguise quite a lot of equipment borrowed from the guts of arithmetic: elliptic PDE concept, complicated differential geometry, algebraic geometry in attribute $p$, cohomological and sheaf-theoretic equipment, deformation thought of advanced forms, Calabi-Yau manifolds, singularity concept, and so forth. a unique attempt has been made to process some of the topics from their such a lot normal beginning issues. all the 3 chapters is supplemented with a close creation and diverse references. The reader will locate designated statements of a range of of open difficulties which have been the topic of lively study in recent times. The reader must have a few familiarity with differential and algebraic geometry, with different necessities various through bankruptcy. The ebook is acceptable as an accompaniment to a moment direction in algebraic geometry.

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