Cohomology Groups, Currents and Dx -Schemes on ∂ -Cohomology

Francisco Bulnes; Sergei Fominko

Francisco Bulnes Professor and Head of Department of Research in Mathematics and Engineering Federal Highway Mexico-Cuautla, Tlapala "La Candelaria", Chalco State of Mexico, P. C. 56641
Sergei Fominko Research Mathematics Department, PreCarpathian University, Ukraine

Keywords: ∂ -cohomology, Cech cohomology, Cohomology Groups, Currents, Dx - Schemes, Radon Transform, Resolution of Acyclic Sheaves.

Abstract

We consider some cohomology groups lemmas as given by Poincaré and Dolbeault-Grothendieck, to establish the De Rham and Dolbeault theorems through currents, and after to be applied to define currents on Dolbeault cohomology. One advantage of this application of currents is the commutation between differential operator and current, which will be demonstrated to a complex holomorphic manifold whose co-cycles under a current are complex domains conformed by holomorphic hyperplanes. In the paper are explained wifely these versions and are applied some Dx -schemes to study of complex holomorphic manifolds and its tomography in cycles of co-dimensions 1, and n - q.

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References

[1] Goldberg, S. I, Curvature and Homology, Dover, New York, N. Y., USA, 1998.
[2] R. Godement (1998), Topologie algébrique et théorie des faisceaux, Hermann, Paris, France. MR 0345092.
[3] J-P. Demailly (2011), Analytic Methods in Algebraic Geometry, Université de Grenoble I, Institut Fourier, France.
[4] I. Verkelov, Moduli Spaces, Non-Commutative Geometry and Deformed Differential Categories, Pure and Applied Mathematics Journal. Special
Issue:Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program. Vol. 3, No. 6-2, 2014, pp. 12-19. doi:
10.11648/j.pamj.s.2014030602.13
[5] F. Bulnes, Integral Geometry Methods in the Geometrical Langlands Program, SCIRP, USA, 2016.
[6] S. Fominko, Approaching by DX- Schemes and Jets to Conformal Blocks in Commutative Moduli Schemes, Pure and Applied Mathematics
Journal. Special Issue:Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program. Vol. 3, No. 6-2, 2014, pp. 38-43.
doi: 10.11648/j.pamj.s.2014030602.17
[7] F. Bulnes, Investigación de Curvatura sobre Espacios Homogéneos, Goverment of State of Mexico, SEP, TESCHA, 2010.
[8] Kobayashi, K. and Nomizu, K. (1969) Foundations of Differential Geometry. Vol. 2, Wiley and Sons, New York.
[9] S. G. Gindikin, G. M. Henkin, Integral geometry for cohomology in q - linear concave domains in, n-P Funcional Anal i Prilozen 12 (1978).
[10] Gelfand, I M and Shilov, Georgi E and Graev, M I and Vilenkin, N Y and Pyatetskii-Shapiro, I I, Generalized Functions, Vol. 5,
Academic Press, New York, N. Y., 1952.
[11] P. Mikusiński and A. Zayed, The Radon Transform of Boehmians, Proceedings of the American Mathematical Society, AMS, Vol. 118, No.
2 (Jun., 1993), pp. 561-570, https://doi.org/10.2307/2160339
[12] F. Bulnes, Radon Transform and the Curvature of a Universe, Postgraduate Thesis, Faculty of Sciences, UNAM, Mexico, 2001.
[13] Bulnes, F., Stropovsvky, Y. and Rabinovich, I. (2017) Curvature Energy and Their Spectrum in the Spinor-Twistor Framework: Torsion as
Indicium of Gravitational Waves. Journal of Modern Physics, 8, 1723-1736. doi: 10.4236/jmp.2017.810101.
[14] F. Bulnes, Dual Representation of the Curvature in a Hilbert Space: Curvature and Integral Transforms, JP Journal of Geometry and
Topology, Vol. 26, (1), pp39 – 51. http://dx.doi.org/10.17654/GT026010039
[15] F. Bulnes, “Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory,”
Journal of Mathematics and Systems Science, Vol. 3 (10) 2013, pp491-507.