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Hodge idea originated as an software of harmonic idea to the learn of the geometry of compact complicated manifolds. the guidelines have proved to be fairly strong, resulting in essentially very important effects all through algebraic geometry. This publication includes expositions of assorted features of recent Hodge conception. Its goal is to supply the nonexpert reader with an actual inspiration of the present prestige of the topic. the 3 chapters increase particular yet heavily comparable topics: $L^2$ Hodge thought and vanishing theorems; Frobenius and Hodge degeneration; adaptations of Hodge buildings and reflect symmetry.The suggestions hired disguise a variety of tools borrowed from the center of arithmetic: elliptic PDE idea, advanced differential geometry, algebraic geometry in attribute $p$, cohomological and sheaf-theoretic tools, deformation idea of advanced kinds, Calabi-Yau manifolds, singularity concept, and so on. a different attempt has been made to procedure many of the topics from their such a lot normal beginning issues. all the 3 chapters is supplemented with a close creation and various references. The reader will locate certain statements of a number of open difficulties which were the topic of lively learn in recent times. The reader must have a few familiarity with differential and algebraic geometry, with different must haves various through bankruptcy. The e-book is appropriate as an accompaniment to a moment path in algebraic geometry.

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