◆ MIXED HODGE FILTRATIONS

DELIGNE WEIGHT FILTRATION // MONODROMY WEIGHT SPECTRAL // Gr^W PURITY

I. Mixed Hodge Structure

A mixed Hodge structure on a finite-dimensional rational vector space H_Q consists of:

An increasing weight filtration W_0 ⊆ W_1 ⊆ ... ⊆ W_{2n} = H_Q
A decreasing Hodge filtration F^n ⊇ F^{n-1} ⊇ ... ⊇ F^0 = H_C

such that each graded piece Gr^W_k = W_k / W_{k-1} carries a pure Hodge structure of weight k.

II. Deligne's Weight Filtration

For H^n(X) where X is a smooth variety with normal crossing divisor boundary D, Deligne constructs W via the logarithmic de Rham complex:

Ω^p_X(log D) = Ω^p_X ⊗ O_X(*D) with poles along D
W_k H^n = Im(H^n(Ω^{≤k}_X(log D)) → H^n(X\D))

The spectral sequence from W degenerates at E_2. Graded pieces: Gr^W_k H^n carries a pure Hodge structure of weight k.

III. Monodromy Weight Filtration

Given a nilpotent endomorphism N: H → H (monodromy logarithm), the monodromy weight filtration M(N, W) is the unique filtration such that:

N: M_k → M_{k-2} and N^k: Gr^M_{n+k} ≅ Gr^M_{n-k}

This is the Jacobson-Morozov filtration centered at the weight. The weight-monodromy conjecture (proven by Deligne in char 0) asserts purity of Gr^M pieces under the Frobenius action.

IV. Extensions and Variations

Mixed Hodge structures form an abelian category MHS. The extension group Ext^1_MHS classifies variations. The period matrix encodes the Hodge filtration position within the weight-graded lattice.

Ext^1_MHS(Q(0), Q(n)) = C / (2πi)^n Q for n ≥ 1
C446 // DELIGNE MIXED HODGE // BBOBOP VOID MATHEMATICS