Applied Mathematicsematics

Download e-book for kindle: Communications in Mathematical Physics - Volume 251 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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30] by exploiting the homological methods developed in ref. [2]. Note, however, that from the point of view of ref. [3] these planar coverings are “´etale” coordinates on X, so the group cohomology constructions required to work with the various kinds of uniformization coverings just ˇ follow from specializing the Cech formalism to the coverings at hand. Finally, the integration in Eqs. 5) should be replaced by the evaluation of [φ] against an appropriate representative of the fundamental class of X.

44 E. 19) at the level of cohomology. ) We will limit ourselves to observe the following: if US is a good cover of S, and UX a good cover of X which refines π −1 US , the i th direct image Ri π∗ F • of any complex F • on X is the sheaf on S associated to the presheaf V −→ H i Tot Cˇ • (UX ∩ π −1 (V ), F • ) . X (or of C(l)•X ) reduces to computing the direct images of Z(l)X and A•X , the de Rham complex. 20) p+q=i with a representative of the fundamental class [M] of the smooth model M of the fiber of π −1 (V ) ∼ = V × M (as smooth manifolds).

3) 2 D d defined on smooth functions φ : D → R, with the condition that variations dα (φα ) α=0 of φ be zero on ∂D. 1), the first term under the integral sign would not yield a well-defined 2-form on X. 3) which we denote 4 Note that Eq. 4) defines an imaginary form. The reason for this choice will be apparent later. 46 E. Aldrovandi ˇ S[φ] = √ −1 ω0 [φ] . 4) is not globally defined. In general terms, given the choice of a conˇ Rham formal metric ds 2 ∈ CM (X) and a cover UX , one works with the full Cech-de • • ˇ complex C (UX , E X (1)) with respect to UX , rather than with just differential forms.

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