# Opposite filtrations, variations of Hodge structure, and Frobenius modules Chapter

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### abstract

• Let X be a complex manifold. Then, an unpolarized complex variation of Hodge structure (E, , F, ) over X consists of a flat, C complex vector bundle (E, ) over X equipped with a decreasing Hodge filtration F and an increasing filtration such that(1.1)F is holomorphic with respect to , and $$abla left({{F^{p}}} ight) subseteq {F^{{p - 1}}} otimes Omega _{X}^{1}$$;(1.2) is anti-holomorphic with respect to , and $$abla left({{{\bar{Phi}}_{q}}} ight)subseteq {\bar{Phi}_{{q + 1}}} otimes overline {Omega _{X}^{1}}$$;(1.3)$$E = {F^{p}} oplus {\bar{Phi }_{{p - 1}}}$$for each index p (i.e. F is opposite to ).
• Let X be a complex manifold. Then, an unpolarized complex variation of Hodge structure (E, ∇, F, Φ̄) over X consists of a flat, C∞ complex vector bundle (E, ∇) over X equipped with a decreasing Hodge filtration F and an increasing filtration Φ̄ such that(1.1)F is holomorphic with respect to ∇, and $$abla left({{F^{p}}} ight) subseteq {F^{{p - 1}}} otimes Omega _{X}^{1}$$;(1.2)Φ̄ is anti-holomorphic with respect to ∇, and $$abla left({{{\bar{Phi}}_{q}}} ight)subseteq {\bar{Phi}_{{q + 1}}} otimes overline {Omega _{X}^{1}}$$;(1.3)$$E = {F^{p}} oplus {\bar{Phi }_{{p - 1}}}$$for each index p (i.e. F is opposite to Φ̄).

### author list (cited authors)

• Fernandez, J., & Pearlstein, G.

• 2

### complete list of authors

• Fernandez, Javier||Pearlstein, Gregory

### editor list (cited editors)

• Hertling, K., & Marcolli, M.

### Book Title

• Frobenius Manifolds

• January 2004