4 added 4 characters in body

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 caseexamples, 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.

3 improved formatting

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, case,there are tons of beautiful constructions appear even in such an elementary level: Hodge structure

infinitesimal VHS, mixed Mixed Hodge structure, period Period mapping $P^{n,k} : B \to Grass(b^{n,k}, H^n(X, \mathbb C)),$ Picad-Fuchs equation and Picard-Lefschetz monosromy representation $\rho: \pi_1(B, b_0) \to GL(n, \mathbb C)$ etcand so on.

My

But my knowledge about of these topics topic remains too abstract to digest it well. So i am collecting lementary, but enlightning enlightening toy examples. For example, I've worked with the Legendre family of elliptic curve 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 needs wants morebecause . Because in my case, no mixed Hodge structure, no intermidiate Jacobian (by a stupid reason), and more missingHodge structure of weight $\ge$ 2. If you have any other good examples, please tell me. A family of surfaces Good reference will be the most desirable exampleextremly helpful. I also appreciate any helpsuggestion.

2 deleted 27 characters in body

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, tons of beautiful constructions appear even in an elementary level: Hodge structure, mixed Hodge structure, period mapping $P^{n,k} : B \to Grass(b^{n,k}, H^n(X, \mathbb C)),$ Picad-Fuchs equation and Picard-Lefschetz representation $\rho: \pi_1(B, b_0) \to GL(n, \mathbb C)$ etc.

My knowledge about these topics remains too abstract to digest it well. So i am looking for an elementarycollecting lementary, but enlightning examples. For example, I recently work I've worked with the Legendre family of elliptic curve

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

and interpret everythings interpreted everything into a concrete term. And term.(and it was a great pleasure. fantastic)

But i still needs more because in my case, no mixed Hodge structuredoes not appear and there is , no intermidiate Jacobian (by a stupid reasonreason), and more missing. If you have any other good examples, please tell me. A family of surfaces will be the most desirable examplesexample. I also appreciate good regerenceany help.

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