Holomorphic connections on holomorphic bundles on Riemann surfaces

We investigate aspects of holomorphic connections on holomorphic principal bundles over a Riemann surface.

Motivic Hypercohomology Solutions in Field Theory and Applications in H-States

Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\textup{DM}_{\textup {gm}}(k)$,  which is of the motivic objects whose image is under $\textup {Spec}(k)$  that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\mathbb{Q}(q)$,  are obtained when cohomology groups of the isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$,   can be vanished for  $p>\textup{dim}(Y)$.  We observe also the Beilinson-Soul$\acute{e}$ vanishing  conjectures where we have the vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \ \textup{if} \ \ p\leq0,$ and $q>0$,   which confirms the before established. Then survives a hypercohomology $\mathbb{H}^{q}(X,\mathbb{Q})$. Then its objects are in $\textup{Spec(Sm}_{k})$.  Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\Sigma$.  Likewise, will be proved that $\mathrm{H}^{\vee}$,  has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$,  on the general linear group with $k=\mathbb{C}$.  A physics re-interpretation of the superposing, to the dual of the spectrum $\mathrm{H}^{\vee}$,  whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ is the corresponding deformed derived category for densities $\mathrm{h} \in \mathrm{H}$, in quantum field theory.

Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $\varphi ^* E$ is holomorphically trivial for some surjective holomorphic map $\varphi $, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

Cycles Cohomology and Geometrical Correspondences of Derived Categories to Field Equations

The integral geometry methods are the techniques could be the more naturally applied to study of the characterization of the moduli stacks and solution classes (represented cohomologically) obtained under the study of the kernels of the differential operators of the corresponding field theory equations to the space-time. Then through a functorial process a classification of differential operators is obtained through of the co-cycles spaces that are generalized Verma modules to the space-time, characterizing the solutions of the field equations. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic bundles category with a special connection (Deligne connection). Using the classification theorem given by geometrical Langlands correspondences are given various examples on the information that the geometrical invariants and dualities give through moduli problems and Lie groups acting.