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On the $p$ -supports of a holonomic $\mathcal{D}$ -module

Thomas Bitoun

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Journal citation: Inventiones Mathematicae 215(3), 779–818 (2019)
Publication date: 2018-11-26
DOI: 10.1007/s00222-018-0837-6 ↗
arXiv counterpart: arXiv:1012.4081v3
Publication license: CC BY 4.0 ↗

Abstract

The paper studies reductions of a holonomic $\mathcal{D}$ -module in positive characteristic, proves equidimensionality and Lagrangianity of their $p$ -supports for large primes, and proves that the differential-operator Azumaya algebra splits on the regular locus of each support component.

Role in dependence graphs

Proof-critical source

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules

This paper is included only for the following marked statement:

Proposition 2.4.1 · Inventiones Mathematicae pp. 787–788; cotangent Lemma A.0.6 on p. 817Closed-immersion direct image transforms $p$ -support by the cotangent correspondence, and Lagrangianity transfers back to the original smooth variety.

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Open-access Springer version of record · Inventiones Mathematicae 215 (2019), no. 3, 779-818

Thomas Bitoun. On the $p$ -supports of a holonomic $\mathcal D$ -module. Inventiones Mathematicae 215 (2019), no. 3, 779-818.

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Generated August 27, 2026

01Statements3 reported findingsNo statement issues found

The main theorem correctly states that, after shrinking a finite-type arithmetic base, the $p$ -support of every closed fiber of a nonzero holonomic $\mathcal D$ -module is Lagrangian. The equidimensionality, degree and rank bounds, Azumaya-splitting theorem, affine-embedding reduction, and characteristic-zero singular-support corollary retain the hypotheses and generic scope needed by their proofs. One ambient-sheaf symbol in Proposition 6.2.2 is a uniquely repairable typographical error.

Theorem 2.2.1 and Corollary 6.3.1No issue found

The Lagrangianity statements have the correct arithmetic and geometric scope

Printed pages 787 and 811-815 · Theorem 2.2.1, its conclusion, and Corollary 6.3.1 · version of record

Theorem 2.2.1 quantifies over the closed points of a dense open subset of the chosen integral finite-type model, rather than over every reduction. It combines dimension $\dim X$ with vanishing of the cotangent symplectic form on a dense smooth locus, which is exactly Definition A.0.5. Corollary 6.3.1 separately spreads a characteristic-zero holonomic module to such a model and recovers the Lagrangian singular support. No stronger all-primes or model-independent assertion is silently used.

Springer version of record ↗

Proposition 2.4.1Correct · focal dependency verified

The affine-space reduction preserves both support dimension and symplectic vanishing

Printed pages 787-788 and 817 · Proposition 2.4.1 and Lemma A.0.6 · version of record

For a closed immersion $f:X\hookrightarrow\mathbb A_S^m$ , the transfer module is flat on the right over $\mathcal D_{X/S}$ and commutes with base change. In characteristic $p$ , the support identity is $\operatorname{p-supp}(H^0f_{s,+}M_s)=(f_s')_\pi\bigl((f_s')_d^{-1}(\operatorname{p-supp}(M_s))\bigr).$ Lemma A.0.6 then transfers vanishing of the symplectic form across the cotangent correspondence, while Theorem 3.1.1 supplies the dimension statement independently. These are precisely the ingredients needed for the reduction.

Springer version of record ↗

Proposition 6.2.2Suspected typo · uniquely repairable

The ambient sheaf in the intersection is printed in the wrong degree

Printed page 809 · statement and proof of Proposition 6.2.2 · version of record

The proposition defines $I$ as the intersection of $\Omega^1_{Y'}((p-1)D')$ and $j'_*\operatorname{Im}(W^*-C_Y)$ but calls both subsheaves of $j'_*\Omega^2_{Y'}$ . Both intersectands consist of 1-forms, and the proof then applies $d$ to obtain a 2-form. Thus the unique correction is to replace that ambient $j'_*\Omega^2_{Y'}$ by $j'_*\Omega^1_{Y'}$ . The conclusion $d(I)\subseteq\Omega^2_{Y'}(\log D')$ and its proof are unchanged.

Springer version of record ↗

02Proofs5 reported findingsNo proof issues found

The proof architecture closes: duality and Azumaya Morita theory give pure $n$ -dimensional supports; Rees-module estimates give uniform degree and generic-rank bounds; coprime annihilators split the differential-operator Azumaya class; and the bounded-pole, $p$ -curvature, Hilbert-scheme, and logarithmic Hodge arguments force symplectic vanishing. The focal closed-embedding formula was also checked directly in local coordinates, independently of its terminal book citation. No unresolved proof gap was found.

Theorem 3.1.1No issue found in the reviewed proof

The purity argument gives equidimensional $p$ -support of dimension $n$

Printed pages 789-793 · Sections 3.2-3.3 · version of record

Generic freeness and specialization of $\mathcal D$ -module Ext groups preserve the characteristic-zero holonomic vanishing outside degree $n$ . After Frobenius pushforward, the Azumaya splitting cover and Morita equivalence identify these vanishings with the corresponding commutative Ext vanishings on $T^*_{X_s'}$ . The double-Ext purity criterion therefore applies with codimension $n$ , and nonvanishing of the fibers gives a pure, hence equidimensional, support of dimension $n$ .

Springer version of record ↗

Theorems 4.3.3 and 5.1.3-5.1.4No issue found in the reviewed proof

The numerical estimates force splitting on the regular support

Printed pages 793-803 · Sections 4.1-5.1 · version of record

The Rees construction for the center turns a good Bernstein filtration into a projective support whose multiplicity is $e(M_\mu)p^n$ . Intersection-theoretic additivity bounds each component degree and its generic module rank. Since the differential-operator algebra has Azumaya rank $p^{2n}$ , every generic rank is $l_zp^n$ with $1\leq l_z\leq e(M_\mu)$ . After excluding the finitely many primes at most $e(M_\mu)$ , the Azumaya class is annihilated by both $l_z$ and $p^n$ ; their coprimality makes the class zero. The closed-embedding argument then transfers splitting from affine space back to $X$ .

Springer version of record ↗

Proof of Theorem 2.2.1No issue found in the reviewed proof

The logarithmic-pole and Hodge-theoretic conclusion is coherent

Printed pages 805-812 · Sections 6.1-6.3 · version of record

The degree bound leaves only finitely many Hilbert polynomials. Proposition 6.1.4 stratifies the relevant Hilbert schemes so that smooth normal-crossings compactifications and a uniform pole bound spread in families. For residue characteristic larger than that bound, Azumaya splitting puts the restricted canonical 1-form in the image of the $p$ -curvature operator, and Proposition 6.2.2 makes its derivative logarithmic. On the characteristic-zero generic fiber this derivative is exact, the logarithmic de Rham complex computes the open de Rham cohomology, and $E_1$ degeneration injects the logarithmic 2-form term into hypercohomology. It therefore vanishes and spreads back to the positive-characteristic fibers.

Springer version of record ↗

Proposition 2.4.1 transfer-module parentVerified focal dependency

The Borel-et-al. input has an independent local-coordinate verification

Printed pages 787-788 · Proposition 2.4.1, citing Borel et al. equation (4) on page 259 · version of record

In normal coordinates for a smooth closed immersion, the transfer module is locally $M\otimes k[\partial_{y_1},\ldots,\partial_{y_c}]$ , up to the standard canonical-bundle twist. It is therefore free, hence flat, over the tangential differential operators and its formation commutes with base change. In characteristic $p$ , each central $y_j^p$ acts by zero, the tangential central operators retain the original support, and the normal $\partial_{y_j}^p$ are unrestricted. This gives exactly the cotangent-correspondence support formula printed in Proposition 2.4.1. The cited Borel et al. book remains a terminal graph node, but the proof-critical content is discharged by this computation.

Springer version of record ↗

Corollary 6.3.1 and Lemma 6.3.2No issue found in the reviewed proof

The reduction-modulo- $p$ proof of classical involutivity closes

Printed pages 813-815 · Corollary 6.3.1 and Lemma 6.3.2 · version of record

A good filtration spreads the singular support to an arithmetic model, while the center filtration realizes each characteristic- $p$ singular support as the reduced special fiber of a flat degeneration of the Lagrangian $p$ -support. The uniform degree bound places these degenerations in finitely many Hilbert families. Lemma 6.3.2 then carries vanishing of the symplectic form from the characteristic-zero generic fiber to reduced closed fibers using a finite base change, normalization, and a fiberwise dense smooth locus. This supplies the claimed characteristic-zero Lagrangian singular support.

On the ppp-supports of a holonomic D\mathcal{D}D-module

Publication metadata

Abstract

Role in dependence graphs

On Lagrangianity of ppp-supports of holonomic DDD-modules and qqq-DDD-modules

Audit summary

Open-access Springer version of record · Inventiones Mathematicae 215 (2019), no. 3, 779-818

The Lagrangianity statements have the correct arithmetic and geometric scope

The affine-space reduction preserves both support dimension and symplectic vanishing

The ambient sheaf in the intersection is printed in the wrong degree

The purity argument gives equidimensional ppp-support of dimension nnn

The numerical estimates force splitting on the regular support

The logarithmic-pole and Hodge-theoretic conclusion is coherent

The Borel-et-al. input has an independent local-coordinate verification

The reduction-modulo-ppp proof of classical involutivity closes

On the $p$ -supports of a holonomic $\mathcal{D}$ -module

On Lagrangianity of $p$ -supports of holonomic $D$ -modules and $q$ - $D$ -modules

The purity argument gives equidimensional $p$ -support of dimension $n$

The reduction-modulo- $p$ proof of classical involutivity closes