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Semistability of the tangent sheaf of singular varieties. (English) Zbl 1379.32010
It is known from the Kobayashi-Hitchin correspondence and the Aubin-Yau theorem on the existence of the Kähler-Einstein metrics that given a compact Kähler manifold \(X\), if its canonical bundle \(K_X\) is ample then the tangent bundle \(T_X\) is polystable with respect to \(K_X\). If \(K_X\) is numerically trivial then the bundle \(T_X\) is polystable with respect to any Kähler class. I. Enoki [Lect. Notes Math. 1339, 118–126 (1988; Zbl 0661.53051)] considered \(X\) with canonical singularities and proved semistability of the tangent sheaf \(\mathcal{T}_X\) in the above situations (we refer to the nicely written article for the definitions).
The main result of the present paper gives further generalizations. Now, one considers a reduced divisor \(D\) such that \((X,D)\) has log canonical singularities. Then it is proven that if \(K_X +D\) is ample then the tangent sheaf \(\mathcal{T}_X (-\log D)\) is polystable with respect to \(K_X +D\). If \(K_X +D\) is trivial then \(\mathcal{T}_X (-\log D)\) is semistable with respect to any Kähler class. Furthermore, if \(D=0\) and \(X\) has Kawamata log terminal singularities then \(\mathcal{T}_X\) is polystable with respect to any Kähler class. As in the proof of Enoki the author uses approximate Kähler-Einstein metrics on the resolution, but he is considering “cuspidal metrics” and also, under weaker assumptions the right hand side of the Monge-Ampère equation is now more degenerate. The proof requires very delicate estimates.

Next, the main result is extended to stable varieties defined as an equidimensional and reduced complex projective variety (say \(X\)) with semi-log canonical singularities such that \(K_X\) is ample. For such \(X\), its normalization \(\nu : X^{\nu} \to X\) and \(\Delta\) the conductor of \(\nu\) the sheaf \(\nu_{\ast}\mathcal{T}_{X^{\nu}}(-\log \Delta)\) is semistable with respect to \(K_X\).
There are two further important results. One gives Bogomolov-type inequalities for projective log canonical pairs. The other one proves generic semipositivity of the sheaf of logarithmic differentials of such pairs with pseudo-effective log canonical class.

32C20 Normal analytic spaces
32Q20 Kähler-Einstein manifolds
32W20 Complex Monge-Ampère operators
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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