Let $\pi: \mathcal{X} \to B$ be a smooth family of compact Kähler manifolds and $\mathcal{V} = R^n\pi_*\mathbb{C}$ be the cohomology sheaf with Hodge filtration $F^p \subset \mathcal{V}$.

Step 1: Gauss-Manin Connection

The Gauss-Manin connection $\nabla: \mathcal{V} \to \Omega^1_B \otimes \mathcal{V}$ is defined by differentiating cohomology classes. For a class $[\alpha] \in H^n(X_t)$: $\nabla_v [\alpha] = \left[\frac{\partial\alpha}{\partial t}\right]$

where $v = \partial/\partial t$ is a tangent vector to $B$ and we choose a representative $\alpha$ of the cohomology class.

Step 2: Hodge Decomposition and Differential

On each fiber $X_t$, we have the Hodge decomposition: $H^n(X_t, \mathbb{C}) = \bigoplus_{p+q=n} H^{p,q}(X_t)$

Let $\alpha \in H^{p,q}(X_t)$ be a harmonic representative. The key observation is that under differentiation: $\frac{\partial \alpha}{\partial t} = \partial\beta + \bar{\partial}\gamma$

for some forms $\beta, \gamma$ of appropriate types.

Step 3: Type Analysis

Decomposing $\partial\alpha/\partial t$ into types, we get components: $\frac{\partial\alpha}{\partial t} = \alpha^{p+1,q-1} + \alpha^{p,q} + \alpha^{p-1,q+1} + \cdots$

The cohomology class $[\partial\alpha/\partial t]$ has contributions from $H^{p+1,q-1}$, $H^{p,q}$, and $H^{p-1,q+1}$.

Step 4: Harmonic Projection

The Gauss-Manin connection applied to $[\alpha] \in H^{p,q}$ gives: $\nabla_v[\alpha] \in H^{p+1,q-1} \oplus H^{p,q} \oplus H^{p-1,q+1}$

But we need to show it lands in $H^{p-1,q+1} \oplus H^{p,q} \oplus \cdots$, i.e., no $H^{p+1,q-1}$ component.

Step 5: Kähler Identities

Using Kähler identities and the fact that $\alpha$ is harmonic: $\Delta\alpha = 0 \implies \partial\alpha = \bar{\partial}\alpha = 0$

The derivative $\partial\alpha/\partial t$ can be written as: $\frac{\partial\alpha}{\partial t} = \frac{\partial g_{i\bar{j}}}{\partial t} \cdot (\text{harmonic correction})$

where $g_{i\bar{j}}$ is the Kähler metric.

Step 6: Infinitesimal Period Relation

The infinitesimal period relation states: $\nabla\Omega \in F^{n-1} \otimes \Omega^1_B$

For $\Omega \in H^{n,0}$, we have: $\nabla_v\Omega \in H^{n-1,1}$

This follows from: $\frac{\partial\Omega}{\partial t} = \eta \wedge \Omega'$

where $\eta \in H^1(T_X)$ is the Kodaira-Spencer class and $\Omega' \in H^{n-1,1}$.

Step 7: General Transversality

For general $p$, the same analysis shows: $\nabla(F^p) \subset F^{p-1} \otimes \Omega^1_B$

The lowering of Hodge type by one reflects the representation theory: differentiating multiplies by coordinates (lowering degree) in the period domain parametrization.

Step 8: Conclusion

Therefore: $\nabla: F^p\mathcal{V} \to \Omega^1_B \otimes F^{p-1}\mathcal{V}$

which is Griffiths transversality.