Abstract
Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
The author thanks Spencer Bloch [Citation3] for sharing his ideas and thanks him for comments on a preliminary version of this paper. He also thanks Jerome William Hoffman, Luc Illusie, Kefeng Liu and Chao Zhang for discussions, and thanks Shiu-Yuen Cheng and Bangming Deng for encouragement.
Notes
1 It should be written as , we omit the letters “” here and in the sequel.
2 This isomorphism can be checked alternatively. For , let l = p in (2.9), then , where the second isomorphism is from Lemma 2.8.
3 When , we have used Angéniol and Lejeune-Jalabert’s method to describe a map from the tangent space of the Hilbert scheme at the point Y to local cohomology in section 3 of [25]. Analogous descriptions were given in [26] (Section 2), where A is a truncated polynomial .
4 Here we use the isomorphism (3.1).
5 It does not depend on the choices of lifting of .
6 We will show that this is a trivial map below.