The Seshadri Constants of Tangent Sheaves on Toric Varieties

In this paper, we investigate the Seshadri constant $\epsilon (X,T_X;p)$ of the tangent sheaf $T_X$ on a complete $\mathbb{Q}$-factorial toric variety $X$. We show that $\epsilon (X,T_X;1)>0$ if and only if the following statement holds true: if $a_1{v}_1+\cdots +a_k{v}_k=0$ where $a_i$’s are positive real numbers and $v_i$’s are primitive generators of some rays in the fan $\mathrm{\Delta }$ that defines $X$, then $k\ge \dim X+1$. Based on the result, we show that a smooth projective toric variety $X$ with $\epsilon (X,T_X;p)>0$ for some $p\in X$ is isomorphic to the projective space, confirming a special case of the conjecture proposed by M. Fulger and T. Murayama.