This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the Noether-Lefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kähler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to Mumford-Tate groups and their associated domains, the Mumford-Tate varieties and generalizations of Shimura varieties.
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 ...
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 ...
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.
[97] P. A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping. Inst. Hautes ́Etudes Sci. Publ. Math., 38:125–180, 1970. [98] A. Grothendieck, Groupes de ...
Under the assumptions and with the notations of that same lemma, we have |F||R 3 L*||a(i,j)||Roe” < LocNeNios NRLeon < co-lockoo, since k > N. Gathering together the information in the above inequalities, we deduce o < R-Mkoko N-LRek ...
Griffiths introduced the period domains and defined the period mappings into period domains in order to study the base space of a family . There exists tautologically the universal variation of Hodge structure on the ...
This volume contains the proceedings of a conference on Hodge Theory and Classical Algebraic Geometry, held May 13-15, 2013, at The Ohio State University, Columbus, OH. Hodge theory is a powerful tool for the study and classification of ...
J. Carlson, S. Miiller-Stach, C. Peters, Period mappings and period domains, Cambridge Studies in Advanced Mathematics, 85. Cambridge University Press, 2003. xvi+430 pp. G. Carlsson, E. Pedersen, Controlled algebra and the Novikov ...
The existence of the H and N + above, given N, is the Jacobson-Morosov theorem. For l ≧ 0, the primitive space is defined by Grl(W(N))prim = kerN l+1 :Grl(W(N)) → Gr−l−2(W(N)). It picks out the sl2-submodule whose highest weight is ...
J. Carlson, S. Müller-Stach, and C. Peters, Period mappings and period domains, Stud. Adv. Math. 85, Cambridge Univ. Press, 2003. E. Cattani, Mixed Hodge structures, compactifications and monodromy weight filtration, Chapter IV in ...