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Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics, Series Number 85)

Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics)

Penned By James Carlson
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Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics, Series Number 85)

Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics)

Penned By James Carlson
4.85 76,445 Ratings
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Editor's Note:
The concept of a period of an elliptic integral goes back to the 18th century. Later Abel, Gauss, Jacobi, Legendre, Weierstrass and others made a systematic study of these integrals. Rephrased in modern terminology, these give a way to encode how the complex structure of a two-torus varies, thereby showing that certain families contain all elliptic curves. Generalizing to higher dimensions resulted in the formulation of the celebrated Hodge conjecture, and in an attempt to solve this, Griffiths generalized the classical notion of period matrix and introduced period maps and period domains which reflect how the complex structure for higher dimensional varieties varies. The basic theory as developed by Griffiths is explained in the first part of the book. Then, in the second part spectral sequences and Koszul complexes are introduced and are used to derive results about cycles on higher dimensional algebraic varieties such as the Noether-Lefschetz theorem and Nori's theorem. Finally, in the third part differential geometric methods are explained leading up to proofs of Arakelov-type theorems, the theorem of the fixed part, the rigidity theorem, and more. Higgs bundles and relations to harmonic maps are discussed, and this leads to striking results such as the fact that compact quotients of certain period domains can never admit a Kahler metric or that certain lattices in classical Lie groups can't occur as the fundamental group of a Kahler manifold.

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Released October 20, 2003
Identifier (ISBN) 1510040
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James Carlson writes contemporary fiction and has always been fascinated by character-driven stories. They hold a degree in English Literature and previously worked as an editor before transitioning to writing their own books. James Carlson lives with their spouse and a very vocal rescue cat.

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"A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel."
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"Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre."
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"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author."
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"Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers."
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"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An absolute must-read for connoisseurs of the genre."
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"Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
Jacob Lewis
Jacob Lewis April 16, 2025
"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
John Harris
John Harris April 26, 2024
"It was adequate. The prose was competent, yet the characters lacked the depth I usually seek. A fair 3 stars."
Mark Turner
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"An acceptable diversion. It verged on repetitive in the second act. A more concise edit would have served it well. Unlikely to reread, but a reasonable expenditure of time."
Michelle Martin
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"Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Edward Martinez
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"A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall."
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Joshua Jackson
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"Exceeded all my expectations. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
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"Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece."
Susan King
Susan King February 19, 2024
"A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I am now a lifelong fan of this author."
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"I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
Brian Lewis
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"A highly sophisticated read. It held my rapt attention, even when certain twists were telegraphed. I would highly recommend this to discerning readers."
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"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
Rebecca Nguyen
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Donna Roberts
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"Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre."
Melissa Nguyen
Melissa Nguyen March 10, 2024
"A decent, if standard, experience. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Unlikely to reread, but a reasonable expenditure of time."
Carol King
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"Took a moment to build, but the payoff was spectacular. It held my rapt attention, even when certain twists were telegraphed. I would highly recommend this to discerning readers."
Sandra Diaz
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"A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. Quite impressive overall."
Nancy Moore
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"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
Nicholas Adams
Nicholas Adams April 30, 2024
"It was adequate. It verged on repetitive in the second act. A more concise edit would have served it well. Unlikely to reread, but a reasonable expenditure of time."
Stephanie Hernandez
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"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre."
Matthew Clark
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"Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
Elizabeth Moore
Elizabeth Moore January 05, 2024
"A very strong entry. A brilliant concept executed with flair. Just a few pacing choices I would change. I would highly recommend this to discerning readers."
Charles Hill
Charles Hill August 30, 2025
"I thoroughly enjoyed the elegance of it. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel."
Carol Torres
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"Not entirely to my refined tastes. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Perfectly average."
Kevin Williams
Kevin Williams April 23, 2025
"Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
Sandra Perez
Sandra Perez May 05, 2025
"It was adequate. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Suitable if your reading queue is empty."
Kevin Clark
Kevin Clark April 27, 2024
"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Dorothy Wilson
Dorothy Wilson December 13, 2025
"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre."
Karen Martinez
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"A very strong entry. The protagonist was compelling, though some subplots felt slightly overshadowed. A wonderful companion for a quiet weekend."
Eric Martinez
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"A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. A very distinguished novel."
Steven Scott
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"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading."
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Daniel Miller August 29, 2025
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Melissa Thompson
Melissa Thompson March 10, 2024
"I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
Robert Adams
Robert Adams June 28, 2024
"A decent, if standard, experience. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. A fair 3 stars."
Susan Harris
Susan Harris June 21, 2024
"Such a rich narrative. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author."
Sharon Lopez
Sharon Lopez May 08, 2024
"A highly sophisticated read. The middle had a deliberate pace, but the climax was phenomenal. A very distinguished novel."
Robert White
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"A decent, if standard, experience. The prose was competent, yet the characters lacked the depth I usually seek. A fair 3 stars."
Elizabeth  Davis
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"Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Kenneth Martin
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"A highly sophisticated read. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers."
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