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H . h. ± δˇ for the total differential. Thus the two cocycles constructed via the direct method of Sect. 2 and the cup product of metrized bundles define the same class. By direct comparison, the proportionality factor is 1. 2 is now easily explained. h. h. h. we obtain the polarization identity ∨ 4[L, ρ] ∪ [L , ρ ] = [L ⊗ L , ρρ ]2 − [L ⊗ L , ρ/ρ ]2 , where the squares in the right hand side refer to ∪. 2, follow by applying Thm. 1 to the latter identity. 1. Let ρ ∈ CM (X) be a conformal metric.

11). Taking into account that the fij are the logarithms of the transition functions, the corresponding (1, 1) class would be given by the associated canonical connection, see Sect. 2. 4. HD a holomorphic one-form ω such that π0 ω = df . In other words it is the group of those smooth functions f such that ∂f is holomorphic, which amounts to say that such an f itself is harmonic. 4. Remarks on the cup product f ∪ g. It is convenient to consider the case of the cup product of two invertible functions f and g in various complexes in more detail, and to introduce some related notions we shall need later.

H . is the cone of the map ρ1 − , where ρ1 : Z(1)•D → R(1)•D , and : F 1A•X ∩ σ 2 E •X (1) → R(1)•D . By unraveling the structure of all the cones involved we have: Z(1)X −−−−−→ 1 X ⊕ OX ı⊕π −−−−−→ 0 2 X ⊕ 1X ı⊕π −−−−−→ 3 X 0 F 1A1X ⊕ E 0X −−−−−→ F 1A2X ⊕ E 1X ⊕0 ⊕ 2X ı⊕π −−−−−→ · · · 0 −−−−−→ F 1A3X ⊕ E 2X ⊕0 −−−−−→ · · · F 1A2X ∩ E 2X (1) −−−−−→ F 1A3X ∩ E 3X (1) −−−−−→ · · · . 13) for ξi ∈ F 1A1X (Ui ), σi ∈ E 0X (Ui ) and ηi ∈ (F 1A2X ∩ E 2X (1))(Ui ). 13), note that each entry is an element of the object in the corresponding position in the left 3 × 3 part of the previous diagram.