Toric Vector Bundles, Concentration Conditions, and Schur Log-Concavity

In the first part (Chapter 3), we give a simple combinatorial proof of the toric version of Mori's theorem that the only smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$. In the second part (Chapter 4), we define a new notion of affine subspace concentration conditions for lattice polytopes and prove that these conditions hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle with the extension class $c_1(\mathcal{T}_X)$ on Fano toric varieties. In the third part (Chapter 5), we consider slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(\mathcal{T}_X)$ on Picard-rank-$1$ Fano varieties. In cases where the index divides the dimension or the dimension plus one, we show that stability of the tangent bundle implies (semi)stability of the canonical extension. One consequence of our result is that the canonical extensions on moduli spaces of stable vector bundles with a fixed determinant on a curve are at least semistable, and stable in some cases. In the last part (Chapter 6), we consider derived polynomials $s_{\lambda}{(i)}$ $i=0,\ldots,|\lambda|$ associated to Schur polynomials $s_{\lambda}$ defined by the rule $$s_{\lambda}(x_1+t,\ldots,x_n+t) = \sum_i s_{\lambda}^{(i)}(x_1,\ldots,x_n) t^i.$$ We conjecture that $$(s_{\lambda}^{(i)})^2 - s_{\lambda}^{(i-1)} s_{\lambda}^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $\lambda = (k^\ell)$, for hooks $\lambda = (k, 1^{\ell -1})$, and when $\lambda = (k,k,1)$ or $\lambda = (3,2^{k-1})$.