000			02044 a2200217 4500
999			_c54509 _d54509
008			210325b ||||| |||| 00| 0 eng d
020			_a9781316639566
082			_a516.35 _bCAR
100			_aCarlson, James,
245			_aPeriod mappings and period domains.
250			_a2nd ed.
260			_bCambridge University Press, _c2018. _aCambridge:
300			_axiv, 562 p. : ill. ; _bpb, _c24 cm.
365			_aGBP _b40.99
504			_aIncludes Biobliography.
520			_aThe 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.
650			_aGeometry
650			_aMathematics
650			_aTopology
942			_2ddc _cTD