The Seshadri constants of tangent sheaves on toric varieties

Abstract

In this paper, we investigate the Seshadri constant $\epsilon (X,T_X;p)$ of the tangent sheaf $T_X$ of a proper $Q$-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 each $a_i$ is a positive real number and each $v_i$ is the primitive generator of some ray in the fan Δ 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.