# A blow up result for a fractionally damped wave equation by Tatar N.

By Tatar N.

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Zn ). Clearly, in what above, by “canonical,” we mean naturally induced by the chosen orientation on the complex line (real plane). 1. Let (U ; z) be a chart in the holomorphic atlas of X, x ∈ U. The real 2n-form oU := dx1 ∧ dy1 ∧ . . 18) is nowhere vanishing on U and defines an orientation for U. One checks that oU = (i/2)n dz1 ∧ dz 1 ∧ . . ∧ dzn ∧ dz n and that (n−1)n oU = (i/2)n (−1) 2 dz1 ∧ . . ∧ dzn ∧ dz 1 ∧ . . ∧ dz n . Let (U ; z ) be another chart around x. We have ∂zj (z(x))|| > 0.

Xn , ∂y1 , . . , ∂yn . Note that TX (R) ⊆ TX (C) via the natural real linear map v → v⊗1. Set ∂zj := 12 (∂xj − i ∂yj ), ∂zk := 12 (∂xj + i ∂yj ). 3) ∂yj = i (∂zj − ∂zj ). 4) Clearly, we have ∂xj = ∂zj + ∂zj , We have TX,x (C) = C ∂z1 , . . , ∂zn , ∂z1 , . . , ∂zn . January 29, 2007 22:22 World Scientific Book - 9in x 6in test Complex manifolds 29 In general, a smooth change of coordinates (x, y) −→ (x , y ) does not leave invariant the two subspaces R {∂xj } and R {∂yj } of TX,x (R). A local linear change of coordinates zj = k Ajk zk (this is the key case to check when dealing with this kind of questions) produces a change of basis in TX,x (C) : (A−1 )kj ∂zk , ∂zj = (A−1 )kj ∂zk .

Can be seen using J via the eigenspace decomposition of TX (C) with respect to J ⊗ IdC . 5): VR ⊆ VR ⊗R C = V ⊕ V where VR is the real vector space underlying V, V and VR ⊗R C has the conjugation operation. 6 and starts with a V as above and considers V ⊕ V instead. We mention this equivalence in view of the use of Hermitean metrics on TX . These metrics are defined as special tensors in TX∗ ⊗C TX∗ . 16) we can view the ∗ ∗ tensor h as an element of TX∗ ⊗C TX∗ ⊆ TX (C) ⊗C TX (C). This is also convenient in view of the use of the real alternating form associated with a Hermitean metric which can then be viewed as a real ∗ element of Λ2C (TX (C)).