User:MatXrg/sandbox

= Definition = Let $$E$$ be a holomorphic vector bundle over a complex manifold $$X$$. Let $$ \mathcal{E}_{\bullet}\, : \, 0 = E_0 \subset E_1 \subset \cdots \subset E_n = E \, $$ be a filtration of $$E$$ by holomorphic subbundles of $$E$$. Let $$ D : E \longrightarrow E \otimes_{\mathcal{O}_X} \Omega_X^1 $$ be a holomorphic connection $$E$$. Then the filtration $$\mathcal{E}_\bullet$$ is said to be Griffiths transversal with respect to the holomorphic connection $$D$$ if it satisfies $$D(E_i) \subseteq E_{i+1} \otimes_{\mathcal{O}_X}\Omega_X^1$$, for all $$1 \leq i \leq n-1$$.