arXiv subjects

Sarjick Bakshi

Publications and source records attributed to Sarjick Bakshi.

6 recordsLinked to original sources

arXiv2026

On Generalized Milnor Manifolds and Their Topological Complexity

We introduce generalized Milnor manifolds (GMM), extending the classical Milnor manifolds over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. We compute their integral cohomology algebras in the complex and quaternionic cases and their mod-$2$ cohomology algebras in the real case. We compare GMM with partial flag manifolds, investigate when they are homotopy equivalent, and obtain a necessary and sufficient condition in the complex and quaternionic cases. We further prove that complex GMM are Kähler manifolds. As an application, we determine their higher topological complexities, obtaining exact values in the complex and quaternionic cases and bounds in the real case. Along the way, for a fibre bundle satisfying the Leray--Hirsch hypothesis, we establish a lower bound for the zero-divisor cup-length of the total space in terms of base and fibre.

math.AT↗

arXiv2026

$g$-vectors and $DT$-$F$-polynomials for Grassmannians

We review $\mathrm{Hom}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen--King--Su's Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of the Plücker coordinates with respect to the triangular initial seed and 2) we express the $F$-polynomials associated with the Donaldson--Thomas transformation in terms of $3$-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.

math.RT↗

arXiv2026

Morphisms from projective spaces to flags of minimal parabolic subgroups

We classify nonconstant morphisms $\mathbb{P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we show that there is no nonconstant morphism $\mathbb{P}^2 \to G/B$; for minimal parabolic subgroup $P_{α_i}$, there are no nonconstant morphisms $\mathbb{P}^3 \to G/P_{α_i}$ when $i \in \{1, n-1\}$, while such morphisms exist for $1 < i < n-1$; and, after correcting an earlier error (pointed out by Yanjie Li), we give an elementary proof that there is no nonconstant morphism $\mathbb{P}^4 \to G/P_{α_i}$ for any minimal parabolic subgroup. The proofs are elementary and cohomological.

math.AG↗

arXiv2021

Smooth torus quotients of Richardson varieties in the Grassmannian

Let $k$ and $n$ be positive coprime integers with $k<n$. Let $T$ denote the subgroup of diagonal matrices in $SL(n,\mathbb{C})$. We study the GIT quotient of Richardson varieties $X^v_w$ in the Grassmannian $\mathrm{Gr}_{k,n}$ by $T$ with respect to a $T$-linearised line bundle $\cal{L}$ corresponding to the Plücker embedding. We give necessary and sufficient combinatorial conditions for the quotient variety $T \backslash\mkern-6mu\backslash (X_w^v)^{ss}_T({\cal L})$ to be smooth.

math.AG↗

arXiv2019

Smooth torus quotients of Schubert varieties in the Grassmannian

Let $r < n$ be positive integers and further suppose $r$ and $n$ are coprime. We study the GIT quotient of Schubert varieties $X(w)$ in the Grassmannian $G_{r,n}$, admitting semistable points for the action of $T$ with respect to the $T$-linearized line bundle ${\cal L}(nω_r)$. We give necessary and sufficient combinatorial conditions for the GIT quotient $T\backslash\mkern-6mu\backslash X(w)^{ss}_{T}({\cal L}(nω_r))$ to be smooth.

math.AG↗

arXiv2019

Torus quotients of Richardson varieties in the Grassmannian

We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus $T$, with respect to the $T$-linearized line bundle ${\cal L}(n ω_r)$ and show that this is smooth when $gcd(r,n)=1$. When $n=7$ and $r=3$ we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This builds on previous work by Kumar \cite{kumar2008descent}, Kannan and Sardar \cite{kannan2009torusA}, Kannan and Pattanayak \cite{kannan2009torusB}, and recent work of Kannan et al \cite{kannan2018torus}. It is known that the GIT quotient of $G_{2,n}$ is projectively normal. We give a different combinatorial proof.

math.RT↗