arXiv subjects
Publications and source records attributed to Sohaib Khalid.
Let $\pi:(X,L)\rightarrow \mathbb D^*$ be the Fermat family of cubic curves in $\mathbb P^2$. For each $k\geq 1$, we construct a valuatively independent basis for $H^0(X,L^k)$. The construction uses a canonical cost function determined by a Hessian structure on the essential skeleton $\op{Sk}(X,\pi)$.
Generalised Monge-Amp\`ere equations form a large class of PDE including Donaldson's J-equation, inverse Hessian equations, some supercritical deformed Hermitian-Yang Mills equations, and some Z-critical equations. Solvability of these equations is characterised by numerical criteria involving intersection numbers over all subvarieties, and in this paper, we aim to characterise algebraically what happens when these nonlinear Nakai-Moishezon type criteria fail. As a main result, we show that under mild positivity assumptions, there is a finite number of subvarieties violating the Nakai type criterion, and such subvarieties are moreover rigid in a suitable sense. This gives first effective solvability criteria for these families of PDE, thus improving on work of Gao Chen, Datar-Pingali, Song and Fang-Ma, and provides first existence results in higher dimension of compact K\"ahler manifolds exhibiting a natural PDE analog of Bridgeland's locally finite wall-chamber decomposition.
We show that on any compact K\"ahler surface existence of solutions to the Z-critical equation can be characterized using a finite number of effective conditions, where the number of conditions is bounded above by the Picard number of the surface.This leads to a first PDE analogue of the locally finite wall-chamber decomposition in Bridgeland stability. As an application we characterize optimally destabilizing curves for Donaldson's J-equation and the deformed Hermitian Yang-Mills equation, prove a non-existence result for optimally destabilizing test configurations for uniform J-stability, and remark on improvements to convergence results for certain geometric flows.
The zeroth-order general Randić index (usually denoted by $R_α^{0}$) and variable sum exdeg index (denoted by $SEI_{a}$) of a graph $G$ are defined as $R_α^{0}(G)= \sum_{v\in V(G)} (d_{v})^α$ and $SEI_{a}(G)= \sum_{v\in V(G)}d_{v}a^{d_{v}}$ where $d_{v}$ is degree of the vertex $v\in V(G)$, $a$ is a positive real number different from 1 and $α$ is a real number other than $0$ and $1$. A segment of a tree is a path $P$, whose terminal vertices are branching or pendent, and all non-terminal vertices (if exist) of $P$ have degree 2. For $n\ge6$, let $\mathbb{PT}_{n,n_1}$, $\mathbb{ST}_{n,k}$, $\mathbb{BT}_{n,b}$ be the collections of all $n$-vertex trees having $n_1$ pendent vertices, $k$ segments, $b$ branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections $\mathbb{PT}_{n,n_1}$, $\mathbb{ST}_{n,k}$, $\mathbb{BT}_{n,b}$. The obtained extremal trees for the collection $\mathbb{ST}_{n,k}$ are also extremal trees for the collection of all $n$-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree $T$ can be determined from the number of vertices of $T$ with degree 2 and vise versa).