Chern character, infinitesimal Abel-Jacobi map and semi-regularity map

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.

KEYWORDS:

• Chern character
• Hochschild homology
• semi-regularity map

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 14C25

Acknowledgments

The author thanks Spencer Bloch [3] 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 $H{H}_0(\text{Spec}(R{\otimes }_{k}A)\text{on}V(J))$, we omit the letters “$\text{Spec}$” here and in the sequel.

2 This isomorphism can be checked alternatively. For $U_i=\text{Spec}(R)$, let l = p in (2.9), then ${\mathbb{H}}_{V(J)}^0(R,H{H}^{(p)}(R\otimes A))=H_J^p(R,H{H}_p^{(p)}(R{\otimes }_{k}A))=H_J^p(R,{\Omega }_{R\otimes A/k}^p)$, where the second isomorphism is from Lemma 2.8.

3 When $A=k[\epsilon ]$, we have used Angéniol and Lejeune-Jalabert’s method to describe a map from the tangent space ${\mathrm{T}}_Y{\text{Hilb}}^p(X)$ 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 $k[t]/(t^j)$.

4 Here we use the isomorphism (3.1).

5 It does not depend on the choices of lifting of $f_{i1}^A,\dots ,f_{\mathit{\text{ip}}}^A$.

6 We will show that this is a trivial map below.