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Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$.

Then, we get local systems $R^i \phi_* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$

In this case,there are tons of beautiful constructions even in such an elementary level:

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.

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

{$y^2=x(x-1)(x-\lambda)$} $ \to $ {$\mathbb C -(0,1)$}

and interpreted everything into a concrete term.(and it was fantastic)

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.

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Have you thought of trying hyperelliptic curves, their products (possibly with diagonals removed), or Jacobians? – S. Carnahan Apr 27 '12 at 3:03
I've tried hyperelliptic curve case already.(I think it is not so different from the Legendre family) But I should try their product surface. – Choa Apr 27 '12 at 3:15
up vote 2 down vote accepted

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.

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Carlson-Muller Stach-Peters is simply great. Full of motivation, inspiring pictures and detailed explaination. They treated my toy example in more depth! It would be a great companion of Voisin's book! Thank you so much. – Choa Apr 27 '12 at 14:28
Griffiths' language is slightly old, but they are great too. – Choa Apr 27 '12 at 14:31

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.

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.

I have learned a lot from the paper "Braid Groups and Hodge Structures" by Curt McMullen: In my opinion, McMullen's papers (on any subject) are absolutely fantastic, and are a great pleasure to read.

In the same direction, some nice examples come from Teichmuller curves. You can check out e.g. the two preprints by Alex Wright: ("Schwarz triangle mappings and Teichmüller curves I: abelian square-tiled surfaces") and ("Schwarz triangle mappings and Teichmüller curves II: the Veech-Ward-Bouw-Möller curves").

Also, some shameless self-advertising: there is
by M. Kontsevich, A. Zorich and me (about square-tiled surfaces) which is mostly expository.

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Buy articles you recommand are also very good and helpful! It improves my previous limited viewpoint. – Choa Apr 27 '12 at 8:50

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