Derived category of toric varieties with Picard number three

We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer

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

Statistics of Moduli Space of vector bundles II

Let $X$ be a smooth irreducible projective curve of genus $g \geq 2$ over a finite field \F_{q} of characteristic $p$ with $q$ elements such that the function field \F_{q}(X) is a geometric Galois extension of the rational function field of degree $N.$ Consider $gcd(n,d)=1$, let $M_{L}(n,d)$ be the moduli space of rank $n$ stable vector bundles over $X$ with fixed determinant isomorphic to a $\mathbb F_q$-rational line bundle $L$. Suppose $N_q (M_L(n,d))$ denotes the cardinality of the set of \F_{q}-rational points of $M_{L}(n,d)$. We give an asymptotic bound of $\log(N_{q}(M_{L}(n,d)) - (n^2-1)(g-1)\log{q})$ for large genus $g,$ depending on $N$. Further, considering this logarithmic difference as a random variable, we prove a central limit theorem over a large family of hyperelliptic curves with uniform probability measure. Further, over the same family of hyperelliptic curves, we study the distribution of \F_{q}-rational points over the moduli space of rank $2$ stable vector bundles with trivial determinant $M^{s}_{\mathcal{O}_{H}}(2,0)$ and it's Seshadri desingularisation ${\widetilde{N}}$ by choosing an appropriate random variable in each case. We also see that the corresponding random variables having standard Gaussian distribution as $g$ and $q$ tends to infinity.Comment: 28 page

Restriction theorems for Higgs principal bundles

AbstractWe prove analogues of Grauert–Mülich and Flennerʼs restriction theorems for semistable principal Higgs bundle over any smooth complex projective variety