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Strong Regularity and Microsupport Estimates for Multi-Microlocalizations of Subanalytic Sheaves

Ryosuke Sakamoto

Abstract

We introduce the notion of strong regularity for subanalytic sheaves and establish estimates for the supports and microsupports of their multi-microlocalizations. As applications, we study subanalytic sheaves of Whit- ney and temperate holomorphic solutions of regular D-modules along an involutive subbundle. In this setting we prove initial value theorems for multi-microlocal objects with growth conditions and division theorems for temperate and Whitney multi-microfunctions. As a consequence, we obtain a multi-microlocal version of Bochner's tube theorem for solution sheaves of strongly asymptotically developable functions.

Strong Regularity and Microsupport Estimates for Multi-Microlocalizations of Subanalytic Sheaves

Abstract

We introduce the notion of strong regularity for subanalytic sheaves and establish estimates for the supports and microsupports of their multi-microlocalizations. As applications, we study subanalytic sheaves of Whit- ney and temperate holomorphic solutions of regular D-modules along an involutive subbundle. In this setting we prove initial value theorems for multi-microlocal objects with growth conditions and division theorems for temperate and Whitney multi-microfunctions. As a consequence, we obtain a multi-microlocal version of Bochner's tube theorem for solution sheaves of strongly asymptotically developable functions.
Paper Structure (11 sections, 26 theorems, 113 equations)

This paper contains 11 sections, 26 theorems, 113 equations.

Table of Contents

  1. Notations
  2. Strongly regular subanalytic sheaves
  3. An estimate of supports and microsupports of multi-microlocalizations for strongly regular subanalytic sheaves
  4. An estimate of supports
  5. Microsupport estimate of multi-microlocalization
  6. Multi-microlocalizations and inverse images
  7. Classical sheaf case
  8. Strongly regular subanalytic sheaf case
  9. Applications to $\mathcal{D}$-modules
  10. Solution of $\mathcal{D}$-modules in complex domains with growth conditions.
  11. The division theorems with growth conditions for regular $\mathcal{D}$-modules

Key Result

Proposition 1.2

Let $F \xrightarrow{f} G \xrightarrow{g} H \xrightarrow{+1}$ be a distinguished triangle in $D^b(k_{X_{sa}})$. Suppose there exists a distinguished triangle of functors from a small filtrant inductive category $I$ to $D^{[a,b]}_{\mathop{\mathrm{\mathbb{R}\text{-}c}}\nolimits}(k_X)$, $F \mapsto F_i$,

Theorems & Definitions (35)

  • Definition 1.1: strongly regular subanalytic sheaves
  • Proposition 1.2
  • Theorem 1.3
  • Claim 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 25 more