Deformation to the normal bundle and blow-ups via derived Weil restrictions

TL;DR

The paper develops a derived analogue of deformation to the normal cone by realizing the deformation to the (derived) normal bundle $\operatorname{D}_{X/Y}$ as a derived Weil restriction along the zero section. It proves an algebraicity theorem for derived Weil restrictions, ensuring $\operatorname{D}_{X/Y}$ is Artin when $f:X\to Y$ is locally of finite type, and extends these techniques to derived blow-ups, providing a universal, functorial picture of exceptional divisors and their projectivizations. The framework yields broad applications to virtual fundamental classes, virtual pull-backs, localization, coherent duality, and microlocalization, and it clarifies relationships between deformation theory, Kontsevich–Behrend-type invariants, and shifted symplectic geometry. The results enable a unified treatment of virtual structures in derived geometry and open pathways to generalized versions in analytic and other contexts, with a coherent treatment of stability, facilitations for stabilizer reduction, and a robust algebraic backbone for derived moduli problems.

Abstract

We develop an analogue of the deformation to the normal cone in the context of derived algebraic geometry. This provides any given morphism of derived stacks with a degeneration to the zero section of its normal bundle (i.e., its 1-shifted relative tangent bundle). The construction is realized via the derived Weil restriction along the zero section of the affine line. We prove a general algebraicity theorem for derived Weil restrictions along finite but possibly non-flat morphisms. As an application of the theory, we study derived blow-ups along arbitrary closed centres, generalizing previous works of the authors in the quasi-smooth case.

Theorems & Definitions (162)

• Theorem 1
• Corollary 2
• Theorem 3
• Corollary 4
• Conjecture 5
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• Corollary 2.4