motivating examples of family of Hodge structure - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:44:29Z http://mathoverflow.net/feeds/question/95314 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95314/motivating-examples-of-family-of-hodge-structure motivating examples of family of Hodge structure Choa 2012-04-27T00:21:00Z 2012-04-27T11:43:50Z <p>Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$.</p> <p>Then, we get local systems $R^i \phi_* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$</p> <p>In this case,there are tons of beautiful constructions even in such an elementary level: </p> <p>infinitesimal VHS, Mixed Hodge structure, Period mapping $P^{n,k} : B \to Grass(b^{n,k}, H^n(X, \mathbb C)),$ Picard-Lefschetz monosromy representation $\rho: \pi_1(B, b_0) \to GL(n, \mathbb C)$ and so on.</p> <p>But my knowledge of these topic remains too abstract to digest it well. So i am collecting enlightening toy examples. For example, I've worked with the Legendre family of elliptic curves</p> <p>{$y^2=x(x-1)(x-\lambda)$} $\to$ {$\mathbb C -(0,1)$}</p> <p>and interpreted everything into a concrete term.(and it was fantastic)</p> <p>But i still wants more. Because in my examples, no mixed Hodge structure, no Hodge structure of weight $\ge$ 2. If you have any other good examples, please tell me. Good reference will be extremly helpful. I also appreciate any suggestion. </p> http://mathoverflow.net/questions/95314/motivating-examples-of-family-of-hodge-structure/95321#95321 Answer by Alex Eskin for motivating examples of family of Hodge structure Alex Eskin 2012-04-27T04:58:38Z 2012-04-27T07:05:35Z <p>Sorry, I misread the question: I thought you wanted examples with no mixed Hodge structure and no Hodge structure of weight $\geq 2$. I tried to delete this post, but it did not work. </p> <p>I am definitely not an algebraic geometer, but I was recently forced to deal with some of the structures you mentioned in some very simple settings. </p> <p>I have learned a lot from the paper "Braid Groups and Hodge Structures" by Curt McMullen: <a href="http://www.math.harvard.edu/~ctm/papers/home/text/papers/bn/bn.pdf" rel="nofollow">http://www.math.harvard.edu/~ctm/papers/home/text/papers/bn/bn.pdf</a>. In my opinion, McMullen's papers (on any subject) are absolutely fantastic, and are a great pleasure to read. </p> <p>In the same direction, some nice examples come from Teichmuller curves. You can check out e.g. the two preprints by Alex Wright: <a href="http://arxiv.org/abs/1203.2683" rel="nofollow">http://arxiv.org/abs/1203.2683</a> ("Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces") and <a href="http://arxiv.org/abs/1203.2685" rel="nofollow">http://arxiv.org/abs/1203.2685</a> ("Schwarz triangle mappings and Teichmüller curves II: the Veech-Ward-Bouw-Möller curves"). </p> <p>Also, some shameless self-advertising: there is <a href="http://arxiv.org/abs/1112.5872" rel="nofollow">http://arxiv.org/abs/1112.5872</a><br> by M. Kontsevich, A. Zorich and me (about square-tiled surfaces) which is mostly expository. </p> http://mathoverflow.net/questions/95314/motivating-examples-of-family-of-hodge-structure/95351#95351 Answer by Donu Arapura for motivating examples of family of Hodge structure Donu Arapura 2012-04-27T11:43:50Z 2012-04-27T11:43:50Z <p>I guess the implied question is: what are good references containing explicit calculations of variations of Hodge structure etc.? I might suggest taking a look at Griffiths' early pioneering papers "On periods of certain rational integrals I, II" Annals 1969, and "Periods of integrals on algebraic manifolds III" IHES 1970. These papers contain a large number of explicit calculations on VHS and intermediate Jacobians for things like hypersurfaces in projective space. Regarding (variations of) mixed Hodge structures, take a look at the books by Carlson-Müller Stach-Peters, Peters-Steenbrink, and Voisin.</p>