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If $X \to S$ is a family of compact Kahler manifolds, then parallel transport with respect to the Gauss-Manin connection on the relative cohomology bundle does not respect the Hodge filtration, e.g. a horizontal relative one-form whose restriction to a given fiber is holomorphic need not be holomorphic when restricted to another fiber.

It's a theorem of Griffiths that at least one has the weaker condition $$ \nabla \text{Fil}^i\mathcal{H}^*_{\text{dR}}(X/S) \subset \text{Fil}^{i-1}\mathcal{H}^*_{\text{dR}}(X/S) \otimes \Omega_S. $$

I noticed in the Wikipedia article that this condition is actually built into the definition of an abstract variation of Hodge structures. Why is this? I assume that there is an application of Griffiths transversality that motivates this?

Related question: why is this result called "transversality?"

Michael Albanese
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    This question screams "MathOverflow" to me. Not only will most members of this site not be able to answer this question, most will not even be familiar with the objects and terminology the question concerns. – Pete L. Clark May 09 '13 at 21:51
  • This really should become a Mathoverflow question! To motivate this let me give a short answer; over there, someone will be able to expand it much better than I could. In the motivating situation of deformations of compact Kähler manifolds, Griffiths transversality reflects that the (differential) of the local period map behaves certainly well. In this setting, it's easy to see why this behaviour translates to Griffiths transversality and, (somehow,) it turns out that requiring it abstractly is good enough for the general theory. – Ben May 11 '13 at 11:50

1 Answers1

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Suppose you have a filtered vector bundle on a space $X$, equipped with a flat (i.e. integrable) connection (which does not necessarily preserve the filtration). If $X$ is simply connected, then we can use the flat connection to identify the fibres of the bundle, and so put ourselves in the situation of having a fixed vector space with a varying filtration, parameterized by $X$.

The space of filtrations will be some partial flag variety $\overline{D}$ (the reason for the overline will be clear in a moment), and so we get a map $X \to \overline{D}$. If our filtrations satisfy some additional conditions like Hodge symmetry, polarizability, etc., then this map will actually land in some open subdomain $D$ of $\overline{D}$, called a period domain.

Finally, if $X$ is not simply connected, we can pull-everything back to its universal cover and apply the above story. The monodromy action of $\pi_1(X)$ on the fibre of our bundle-with-flat-connection will give a map $\pi_1(X) \to \Gamma$, where $\Gamma$ is a discrete group acting on $D$.

So altogether we get the period map $$X \to \Gamma \backslash D.$$

For example, if $X = \mathbb P^1 \setminus \{0,1,\infty\}$ and our bundle-with-connection is the variation of Hodge structure coming from the $H^1$ of the Legendre family of elliptic curves $y^2 = x(x-1)(x-\lambda)$, then $D$ will be the upper half-plane $\mathcal H$, the group $\Gamma$ will be the congruence subgroup $\Gamma(2)$ in $SL(2,\mathbb Z)$, and the period map will be the standard isomrphism $$\mathbb P^1 \setminus \{0,1,\infty\} \cong \Gamma(2) \backslash \mathcal H.$$

Now the reason Griffiths introduced the concept of variation of Hodge structures is to use them as a tool to study such period maps for more general families of varieties, and use them to deduce geometric facts (about cycles and so on in the members of the family). So he wanted a set of axioms that captured the key properties of the situation. And Griffiths transversality is one of these key properties.

What it says is that the period map $X \to \Gamma \backslash D$ is constrained in a certain way. There are different ways to phrase this constraint (all just rephrasings of Griffiths transversality): one way is differential geometric in nature, and literally says that the period map has to be transverse to a certain distribution on $\Gamma \backslash D$. (Here distribution is in the sense of differential topology.) This is where the name comes from.

Matt E
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