On equivariant Serre problem for principal bundles

Let $E_G$ be a $\Gamma$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $\Gamma$, where $G$ and $\Gamma$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\Gamma$--space with a dense orbit, and $x_0 \in X$ is a $\Gamma$--fixed point. We show that if $\Gamma$ is reductive, then $E_G$ admits a $\Gamma$--equivariant isomorphism with the product principal $G$--bundle $X \times_{\rho} E_G(x_0)$, where $\rho\,:\, \Gamma \, \longrightarrow\, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety.Comment: References added. To appear in the International Journal of Mathematic

Tannakian classification of equivariant principal bundles on toric varieties

Let $X$ be a complete toric variety equipped with the action of a torus $T$ and $G$ a reductive algebraic group, defined over an algebraically closed field $K$. We introduce the notion of a compatible $\Sigma$--filtered algebra associated to $X$, generalizing the notion of a compatible $\Sigma$--filtered vector space due to Klyachko, where $\Sigma$ denotes the fan of $X$. We combine Klyachko's classification of $T$--equivariant vector bundles on $X$ with Nori's Tannakian approach to principal $G$--bundles, to give an equivalence of categories between $T$--equivariant principal $G$--bundles on $X$ and certain compatible $\Sigma$--filtered algebras associated to $X$, when the characteristic of $K$ is $0$.Comment: 22 pages, revised version, to appear in Transform. Group

A classification of equivariant principal bundles over nonsingular toric varieties

We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.Comment: 14 page

Reviewing the prospect of fermion triplets as dark matter and source of baryon asymmetry in non-standard cosmology

Indirect searches of Dark Matter (DM), in conjugation with `missing track searches' at the collider seem to confine SU(2)$_L$ fermion triplet DM (FTDM) mass within a narrow range around 1 TeV. The canonical picture of the pure FTDM is in tension since it is under-abundant for the said mass range. Several preceding studies have reported that an extra species ($\phi$), redshifts faster than the radiation ($\sim a^{-(4+n)}$ where $n>0$), leads to a faster expanding early Universe by dominating in the energy density with an enhanced Hubble parameter. This has the potential to revive the under-abundant FTDM ($\mathbb{Z}_2$ odd, lightest generation) by causing freeze-out earlier without modifying the interaction strength between DM and thermal bath. On the other hand, although the CP asymmetry produced due to the decay of $\mathbb{Z}_2$ even heavier generations of the triplet remains unaffected, its evolution is greatly affected by the non-standard cosmology. It has been observed through numerical estimations that the minimum mass of the triplet, required to produce sufficient baryon asymmetry of the Universe (BAU), can be lowered up to two orders (compared to the standard cosmology) in this fast expansion scenario. The non-standard parameters $n$ and $T_r$ (a reference temperature below which radiation dominance prevails), which simultaneously control DM abundance as well as the frozen value of BAU, are tightly constrained from the observed experimental values. We have found that $n$ is strictly bounded within the interval $0.4\lesssim n \lesssim 1.8$ where the upper bound is imposed by the BAU constraint whereas the lower bound arises to satisfy the correct DM abundance. It has been noticed that the restriction on $T_r$ is not so stringent as it can vary from sub-GeV to a few tens of GeV.Comment: 40 pages, 10 figures, 1 table, minor changes, version published in JCA

Structural segments and residue propensities in protein-RNA interfaces: Comparison with protein-protein and protein-DNA complexes

The interface of a protein molecule that is involved in binding another protein, DNA or RNA has been characterized in terms of the number of unique secondary structural segments (SSSs), made up of stretches of helix, strand and non-regular (NR) regions. On average 10-11 segments define the protein interface in protein-protein (PP) and protein-DNA (PD) complexes, while the number is higher (14) for protein-RNA (PR) complexes. While the length of helical segments in PP interaction increases with the interface area, this is not the case in PD and PR complexes. The propensities of residues to occur in the three types of secondary structural elements (SSEs) in the interface relative to the corresponding elements in the protein tertiary structures have been calculated. Arg, Lys, Asn, Tyr, His and Gln are preferred residues in PR complexes; in addition, Ser and Thr are also favoured in PD interfaces

On stability of tangent bundle of toric varieties

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.Comment: Revised version. To appear in Proc. Indian Acad. Sci. Math. Sc