Li — On The Stability Of Extensions Of Tangent Sheaves On Kähler–Einstein Fano/Calabi–Yau Pairs (2021)

Let S be a smooth projective variety and $Δ $a simple normal crossing ${\mathbb { Q}}$-divisor with coefficients in (0,~1]. For any ample ${\mathbb { Q}}$-line bundle L over S, we denote by $\mathscr { E} (L)$the extension sheaf of the orbifold tangent sheaf $T_S(-log (Δ ))$by the structure sheaf $\mathcal { O}_S$with the extension class \(c_1(L)\). We prove the following two results: (1)if $-(K_S+Δ )$is ample and $(S, Δ )$is K-semistable, then for any \(\lambda \in {\mathbb { Q}}_{{\)>\(}0}\), the extension sheaf $\mathscr { E} ({λ c_1(-(K_S+Δ ))} )$is slope semistable with respect to \(-(K_S+\Delta )\);(2)if \(K_S+\Delta \equiv 0\), then for any ample ${\mathbb { Q}}$-line bundle L over S, $\mathscr { E} (L)$is slope semistable with respect to L.These results generalize Tian’s result where $-K_S$is ample and $Δ =∅ $. We give two applications of these results. The first is to study a question by Borbon–Spotti about the relationship between local Euler numbers and normalized volumes of log canonical surface singularities. We prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked Riemann surface. In particular we complete the computation of Langer’s local Euler numbers for any line arrangements in \({\mathbb { C}}\textasciicircum 2\). The second application is to derive Miyaoka–Yau-type inequalities on K-semistable log-smooth Fano pairs and Calabi–Yau pairs, which generalize some Chern-number inequalities proved by Song–Wang.