Automorphism and Cohomology II: Complete intersections

TL;DR

The work analyzes automorphisms of smooth complete intersections in projective spaces and their action on cohomology. It combines equivariant deformation theory, the equivariant Kodaira–Spencer map, and infinitesimal Torelli to establish generic triviality of $\mathrm{Aut}_L(X)$ and faithfulness of $\mathrm{Aut}(X)$ on the middle cohomology in characteristic 0, with precise exceptions. In positive characteristic, the authors leverage lifting techniques (Pan–L) to transfer the faithfulness result from characteristic 0, while providing a detailed treatment of the special $(2,2)$-type case and other low-dimensional exceptions. Overall, the paper extends classical results on automorphisms from hypersurfaces to higher codimension complete intersections and clarifies how automorphisms interact with crystalline and étale cohomology across characteristics.

Abstract

We prove that the automorphism group of a general complete intersection $X$ in a projective space is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the cohomology of $X$ faithfully with a few well-understood exceptions.

Theorems & Definitions (39)

• Theorem 1.1: Matsumura and Monsky
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['LEMCIAUTO100']}
• Lemma 2.3