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Analysis

This paper introduces a new class of rigid analytic varieties over a p-adic field that exhibit Poincaré duality for étale cohomology with mod p coefficients. The significance lies in extending Poincaré duality results to a broader class of varieties, including almost proper varieties and p-adic period domains. This has implications for understanding the étale cohomology of these objects, particularly p-adic period domains, and provides a generalization of existing computations.
Reference

The paper shows that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class.

Analysis

This paper presents three key results in the realm of complex geometry, specifically focusing on Kähler-Einstein (KE) varieties and vector bundles. The first result establishes the existence of admissible Hermitian-Yang-Mills (HYM) metrics on slope-stable reflexive sheaves over log terminal KE varieties. The second result connects the Miyaoka-Yau (MY) equality for K-stable varieties with big anti-canonical divisors to the existence of quasi-étale covers from projective space. The third result provides a counterexample regarding semistability of vector bundles, demonstrating that semistability with respect to a nef and big line bundle does not necessarily imply semistability with respect to ample line bundles. These results contribute to the understanding of stability conditions and metric properties in complex geometry.
Reference

If a reflexive sheaf $\mathcal{E}$ on a log terminal Kähler-Einstein variety $(X,ω)$ is slope stable with respect to a singular Kähler-Einstein metric $ω$, then $\mathcal{E}$ admits an $ω$-admissible Hermitian-Yang-Mills metric.