arXiv subjects
Publications and source records attributed to Saurav Bhaumik.
In this article, we compute both the algebraic and the analytic Brauer groups of a homogeneous space under the action of a connected, simply connected, semisimple complex algebraic group, where the stabilizer subgroup is closed and connected.
In density functional theory (DFT) based total energy studies, the van der Waals (vdW) and zero-point vibrational energy (ZPVE) correction terms are included to obtain energy differences between polymorphs. We propose and compute a new correction term to the total energy, due to electron-phonon interactions (EPI). We rely on Allen's general formalism, which goes beyond the Quasi-Harmonic Approximation (QHA), to include the free energy contributions due to quasiparticle interactions. We show that, for semiconductors and insulators, the EPI contributions to the free energies of electrons and phonons are the corresponding zero-point energy contributions. Using an approximate version of Allen's formalism in combination with the Allen-Heine theory for EPI corrections, we calculate the zero-point EPI corrections to the total energy for cubic and hexagonal polytypes of Carbon, Silicon and Silicon Carbide. The EPI corrections alter the energy differences between polytypes. In SiC polytypes, the EPI correction term is more sensitive to crystal structure than the vdW and ZPVE terms and is thus essential in determining their energy differences. It clearly establishes that the cubic SiC-3C is metastable and hexagonal SiC-4H is the stable polytype. Our results are consistent with the experimental results of Kleykamp. Our study enables the inclusion of EPI corrections as a separate term in the free energy expression. This opens the way to go beyond the QHA by including the contribution of EPI on all thermodynamic properties.
Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w=(s_{i_{1}},s_{i_{2}},\ldots, s_{i_{r}})$ of simple reflections in $W,$ let $Z_{w}$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w.$ Let $\widetilde{Z_{w}}:=G\times^{B}Z_{w}$ denote the fibre bundle over $G/B$ with the fibre over $B/B$ is $Z_{w}.$ In this article, we give necessary and sufficient conditions for the varieties $Z_{w}$ and $\widetilde{Z_{w}}$ to be Fano (weak-Fano). We show that a line bundle on $Z_{w}$ is globally generated if and only if it is nef. We show that Picard group $\text{Pic}(\widetilde{Z_{w}})$ is free abelian and we construct a $\mathcal{O}(1)$-basis. We characterize the nef, globally generated, ample and very ample line bundles on $\widetilde{Z_{w}}$ in terms of the $\mathcal{O}(1)$-basis.
Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $ρ:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but we discuss some important situations where the answer is positive. We show that if $G$ is solvable and $ρ:G\to GL(U)$ is a complex representation, then the only universal complex subspace is $U$ itself. We also investigate the question of universality for a Levi subgroup.
Let $G$ be a connected Lie group. In this paper, we study the density of the images of individual power maps $P_k:G\to G:g\mapsto g^k$. We give criteria for the density of $P_k(G)$ in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if ${\rm Reg}(G)$ is the set of regular elements of $G$, then $P_k(G)\cap {\rm Reg}(G)$ is closed in ${\rm Reg}(G)$. On the other hand, the weak exponentiality of $G$ turns out to be equivalent to the density of all the power maps $P_k$. In linear Lie groups, weak exponentiality reduces to the density of $P_2(G)$. We also prove that the density of the image of $P_k$ for $G$ implies the same for any connected full rank subgroup.
Let $GO(2n)$ be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$ of the classifying space $BGO(2n,\mathbb C)$ of the corresponding complex Lie group $GO(2n,\mathbb C)$ in terms of explicit generators and relations. The author of the present note showed that over any algebraically closed field of characteristic not equal to $2$, the smooth-étale cohomology ring $H_{\rm sm-ét}^*(BGO(2n),\mathbb F_2)$ of the classifying algebraic stack $BGO(2n)$ has the same description in terms of generators and relations as the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$. Totaro defined for any reductive group $G$ over a field, the Chow ring $A^*_G$, which is canonically identified with the ring of characteristic classes in the sense of intersection theory, for principal $G$-bundles, locally trivial in étale topology. In this paper, we calculate the Chow group $A^*_{GO(2n)}$ over any field of characteristic different from $2$ in terms of generators and relations.
For a regular pair $(X,Y)$ of schemes of pure codimension 1 on which 2 is invertible, we consider quadric bundles on $X$ which are nondegenerate on $X-Y$, but are minimally degenerate on $Y$. We give a formula for the behaviour of the cohomological invariants (characteristic classes) of the nondegenerate quadric bundle on $X-Y$ under the Gysin boundary map to the etale cohomology of $Y$ with mod 2 coefficients. The results here are the algebro-geometric analogs of topological results for complex bundles proved earlier by Holla and Nitsure, continuing further the algebraization program which was commenced with a recent paper by Bhaumik. We use algebraic stacks and their smooth-etale cohomologies, $A^1$-homotopies and Gabber's absolute purity theorem as algebraic replacements for the topological methods used earlier, such as CW complexes, real homotopies, Riemannian metrics and tubular neighbourhoods. Our results also hold in smooth-etale cohomology for quadric bundles over algebraic stacks on which 2 is invertible.
Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$.
Let GO(2n) be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic not equal to 2. We determine the etale cohomology ring with mod 2 coefficients of the algebraic stack BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring of the classifying space BGO(2n) of the complex Lie group GO(2n) in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are (i) an extension to etale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the G_m fibration BO(2n) to BGO(2n) of algebraic stacks.