## 1. Introduction

The complex analysis development can establish three fundamental ways of modern research with the goal to solve certain conjectures and complete certain theories as the Lie groups representation theory, obtaining of a general cohomology for certain integral operators in the differential equations solution and extend these solutions to the meromorphic context [1].

The three ways are defined for the relations between the complex analysis, with the cohomology with coefficients in holomorphic vector bundles, the hyper-complex analysis in a quaternionic algebra and the unitary representations in the Lie group theory [2].

Likewise, some objects as modular forms can be the key for the multidisciplinary investigations in complex functions without the condition that

The transcendence goes more beyond of the analyticity problems in complex analysis. The study of the complex Riemannian manifolds results relevant in algebraic geometry and strings theories, where complex Riemann surfaces as complex submanifolds can be useful in the determination of integration invariants. Also in the Fröbenius distribution for the determination of integral submanifolds as cycles, whose values are co-cycles in a spectrum of the category

The theory of Riemann surfaces can be applied to the space ^{1}:

In projective spaces, the functions are homogeneous polynomials. These algebro-geometric objects are the sections of a sheaf (one could also say a line bundle in this case). Precisely here arises the re-interpretation of the differential operators through the homogeneous polynomials of the corresponding lines bundles sections of their respective connections [5] ( Figure 1 ).

However, there is other conjecture established years ago, which establish the sentence.

**Conjecture 1.1.** All complex function (analytic function)

This could be obtained in a clear way, without pass for the holomorphicity.

This represents all a joint research program between the hypercomplex analysis and the integral operators cohomology, and for it is established determine an integral operators theory that establish equivalences between cohomology classes on different complex base spaces and the cycles and co-cycles of these under the corresponding integral operators.

Likewise, can be obtained integral representations that are realizations of certain unitary representations of Lie groups such as

However, the isomorphism relations in homogeneous vector bundles are equivalent to the followed in cohomology developed in vector tomography. Likewise, the intertwining integral operators between cohomological classes of both contexts (respective cycles) result be cohomological classes in contexts of lines bundles (or equivalent co-cycles). In this sense a conjecture that can be proved is: “the Penrose transform is a vector Radon transform on homogeneous vector bundles sections in a Riemannian manifold” And considering the re-construction of a Riemannian manifold through cycles, we can consider the following conjecture also.

**Conjecture 1.2.** [2] The twistor transform is the generalization of the Radon transform on lines bundles and the Penrose transform is a specialization of the twistor transform in

Of fact, the twistor transform can be determined for the Penrose transforms pair on

The spaces or cohomology groups of the complex corresponding of sheaves

However, this is not reduced to the obtaining of representations of complex groups. Also the obtaining of an analytic function

which represents a transformation in cycles (lines) of a flag manifold

Other study is the singularities study through varieties considering newly the complex varieties, in a pure algebraic treatment, or using the analyticity as a concept required for contour integrals to define a singularity as a pole of an analytic function.

Likewise in concrete, through an analytic functions study, can be extended and generalized the Cauchy integral as through integrals of contour can be generalized functional in a cohomology of contours (cohomological functional). The Morera’s and Cauchy-Goursat’s theorems can be applied in a sense geometrical more general as cohomological functional of contours whose residue value can be modulo

As a way to extend the meromorphic functions, the Nevanlinna theory searches generalizations of extensions of analytic functions to algebroid functions, holomorphic curves [10], holomorphic maps between complex manifolds of arbitrary dimension, quasi-regular mappings and minimal surfaces [11]. This is being applicable in moduli problems and Shimura varieties.

## 2. Future research

Finally, the future researches in complex analysis will be centered on major research of meromorphicity, in the automorphicity [12, 13], bi-holomorphicity, Riemann-Zeta function, Hodge theory and the complex ideals for improve methods of homogeneous polynomials in the complex vector bundles studied in complex Riemaniann manifolds and their submanifolds, involving singularities as poles or insolated singularities of rational curves. The automorphic forms generalize the Cauchy and Morera’s theorems in an arithmetic sense inside the analytic functions theory and their probable holomorphicity. The automorphic forms are the other basic operations that arise inside the mathematics in the modular forms.

## Notes

- Here , , is a floor function.