# Communications in Mathematical Physics - Volume 251 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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**Example text**

H . becomes just the cone of the morphism Z(2)•D −→ E •X (1)[−1]. h . 2) where the differentials have been written explicitly. 2) and the shift of two positions to the right clearly accounts for the cohomology degree shift from 2 to 4. 1. Let X be a compact Riemann surface of genus g > 1. Let (L, ρ) and (L , ρ ) be two hermitian holomorphic line bundles. h. log ρ, log ρ constructed in Sect. 4. 4 Proof. h. the class of [log ρ, log ρ ] is in H2 (X, E •X (1)) ∼ = H 2 (X, R(1)). Note that for X connected they must coincide up to a proportionality factor, since H 2 (X, R(1)) ∼ = R(1) in this case.

H . h . h . 1 is: Hermitian-Holomorphic Classes Z(2)X −→ OX −→ |π ↓ 1 2 X ⊕ 43 1 X −→ | ı ⊕ π1 ↓ 3 X ⊕ 2 X −→ 4X ⊕ | ı ⊕ π1 ↓ 3 X −→ . . | ı ⊕ π1 ↓ 2 (1) −→ F 2 A4 ⊕ E 3 (1) −→ . . 17) ↑ | ı⊕0 4 (2) −→ . . h . would result in an analogous 5 × 3 table. The degrees (in the cone) start from 0 in the leftmost entry in the top and bottom rows, and from 1 in the center one. 5) plus those arising from the ˇ Cech resolution as explained in Sect. 2. 18) below, where the corresponding bidegrees are explicitly indicated: the first degree is the overall degree in the cone and the second ˇ is the Cech degree.

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. 6) is given by the (evaluation of) the square [TX , ρ] ∪ [TX , ρ] with respect to the cup product in hermitian holomorphic Deligne cohomology. 6). 1. h . the descent equations are explicit and close automatically.