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Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities

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Pith's one-line read This paper proves that spectral Hirzebruch-Milnor classes of local complete intersections vanish exactly outside a window set by the minimal exponent, and uses that to detect higher Du Bois and rational singularities.

desk verdict New lci Hirzebruch–Milnor classes with a real theorem: minimal exponent controls spectral class vanishing, but the key gluing lemma is quoted from an unreviewed preprint. read the letter →

arxiv 2502.06537 v1 pith:TW4ZX3MC submitted 2025-02-10 math.AG

Bradley Dirks , Laurenţiu Maxim , Sebastián Olano This is my paper · ORCID
classification math.AG MSC 14J1732S3532S5014C17
keywords minimalexponentHirzebruch-MilnorclasseslocalcompleteintersectionshigherDuBoissingularitiesrationalHodgespectrumV-filtrationcharacteristic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces Hirzebruch-Milnor type characteristic classes for local complete intersection subvarieties of smooth complex algebraic varieties, using the deformation to the normal cone and the associated specialization functor. Its central claim is that the coefficients of these classes, indexed by a spectral parameter $t$, vanish outside an explicit range controlled by the minimal exponent $ ilde{\alpha}(X)$: above $\dim Y+r-\tilde{\alpha}(X)$ and below $\tilde{\alpha}(X)-r+1$, with converses when the singular locus is projective and the log canonical threshold satisfies $\operatorname{lct}(X)>r-1$. Because the minimal exponent encodes when a local complete intersection has $k$-Du Bois or $k$-rational singularities, these vanishings turn the classes into homological tests for those higher singularity notions. For isolated singularities, the results recover and refine known formulas for the Hodge spectrum.

What carries the argument

The central object is the vanishing-cycle mixed Hodge module $\phi_h Q^H_{\tilde Y}$ attached to the deformation to the normal cone $h:\tilde Y\to \mathbb{C}$, whose special fiber is the normal bundle $C_XY$ of $X$ in $Y$. The spectral Hirzebruch-Milnor class is obtained by applying the spectral Hirzebruch class transformation (the Todd class of a spectral motivic Chern class) to this module and pulling back along the zero section of the normal bundle. A gluing lemma, quoted from an unpublished preprint, identifies the lowest non-vanishing graded de Rham piece of the vanishing-cycle module with a pullback sheaf $\pi^*(F)$ on the normal bundle over the singular locus; this identification is what converts non-vanishing of a coherent sheaf on a projective singular locus into non-vanishing of the corresponding characteristic class coefficient.

What would settle it

Take a codimension $r$ local complete intersection $X\subset Y$ with non-isolated, projective singular locus and $\operatorname{lct}(X)>r-1$, such as a cone over a smooth projective complete intersection, and compute the spectral Hirzebruch-Milnor class via the deformation-to-normal-cone construction; any non-zero coefficient outside the window $[\tilde{\alpha}(X)-r+1,\ \dim Y+r-\tilde{\alpha}(X)]$ would falsify Theorems 4.21 and 4.24. Alternatively, directly test the gluing lemma by computing the lowest Hodge piece of $\phi_h$ for two different regular sequences defining the same ideal and comparing the resulting pullback sheaves; a mismatch would break the converse arguments.

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Extended reading notes

Core claim

The central discovery is that the spectral Hirzebruch-Milnor class $M^{sp}_{t*}(X\subset Y)$ of a codimension $r$ local complete intersection $X$ in a smooth variety $Y$ is a sharp homological shadow of the minimal exponent $\tilde{\alpha}(X)$. Under the hypothesis $\operatorname{lct}(X)>r-1$, the paper proves that the $t^{\alpha}$-coefficient vanishes for all $\alpha>\dim Y+r-\tilde{\alpha}(X)$ and for all $\alpha<\tilde{\alpha}(X)-r+1$; if $\operatorname{Sing}(X)$ is projective, the converses hold, so the vanishing range characterizes the minimal-exponent bound. The proof first establishes a new formula for $\tilde{\alpha}(X)$ as $r-1+\min\{p+\lambda\}$, where the minimum is taken over the lowest non-vanishing graded de Rham piece of the vanishing-cycle mixed Hodge module of the deformation to the normal cone. As applications, the paper shows that the $y$-coefficients of the (unipotent and non-unipotent) Hirzebruch-Milnor classes vanish in specified ranges exactly when $X$ has $k$-Du Bois or $k$-rational singularities, with converses under the same projectivity and log canonical threshold assumptions.

Load-bearing premise

The central equivalence rests on a gluing lemma borrowed from an unreviewed preprint that identifies the first non-vanishing Hodge piece of the vanishing-cycle module with a coherent sheaf pulled back from the singular locus; if that lemma fails for non-isolated singularities, the vanishing theorems and their converses collapse.

Editorial extensions

If this is right

  • For isolated singularities, the Hodge spectrum's smallest non-zero exponent is $\tilde{\alpha}(X)-r+1$ and its largest is $\dim Y-\tilde{\alpha}(X)$, refining earlier spectrum formulas.
  • A local complete intersection with projective singular locus and $\operatorname{lct}(X)>r-1$ is $k$-Du Bois if and only if the non-unipotent coefficients vanish for all $p\geq \dim Y-k$ and the unipotent coefficients vanish for all $p\geq \dim Y+1-k$.
  • Under the same hypotheses, $X$ is $k$-rational if and only if $[M_{y*}(X\subset Y)]_p=0$ for all $p\geq \dim Y-k$, with the additional unipotent vanishing $[M^{\{1\}}_{y*}(X\subset Y)]_{k+1}=0$.
  • Smooth local complete intersections have identically zero Hirzebruch-Milnor classes, while singular ones have a non-zero coefficient of the fundamental class of the singular locus, giving a simple singularity detector.
  • The lci results extend the hypersurface spectral vanishing theorems from codimension one to arbitrary codimension, with the same sharp threshold behavior in terms of the minimal exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof's reliance on Hodge-theoretic duality suggests that the spectral classes satisfy a $t\leftrightarrow t^{-1}$ symmetry with dimension twist, extending the duality formula known for Hodge spectra of isolated singularities; making this symmetry explicit could give a direct bridge between the two vanishing theorems.
  • Because $\operatorname{lct}(X)>r-1$ is automatic for Du Bois lci singularities, the converses apply to the entire Du Bois regime, so the class-level tests are complete precisely where the minimal exponent is governed by Hodge-theoretic vanishing.
  • The new minimal-exponent formula in Theorem 4.7 does not itself involve characteristic classes and could be used independently to compute $\tilde{\alpha}(X)$ from local $V$-filtration data in examples with non-isolated singular locus.
  • The gluing lemma that identifies the lowest Hodge piece with a pullback sheaf may be the natural tool to extend the results to analytic singularities or to other homology theories admitting a Todd-class Riemann-Roch, although the paper notes that compactness alone does not replace projectivity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper defines (spectral) Hirzebruch-Milnor type homology characteristic classes for any local complete intersection subvariety X of a smooth complex algebraic variety Y, using the deformation to the normal cone, Verdier-Saito specialization, and the spectral Hirzebruch class transformation. The main results (Theorems 4.21 and 4.24, stated also as Theorems 1.1 and 1.4) establish vanishing of the spectral classes outside an interval determined by the minimal exponent ~α(X), with converses under the assumptions that Sing(X) is projective and lct(X)>r−1. As applications, the authors obtain class-level characterizations of k-Du Bois and k-rational singularities for lci varieties (Theorems 4.29 and 4.30) and new proofs of formulas for the Hodge spectrum of isolated complete intersection singularities (Corollaries 4.23 and 4.25).

Significance. This is a substantial and well-motivated contribution. It extends the hypersurface theory of Hirzebruch-Milnor classes and their spectral refinements to arbitrary lci varieties, giving global homological refinements of the Hodge spectrum that detect higher Du Bois and rational singularities. The definitions are natural, the machinery (mixed Hodge modules, V-filtrations, specialization) is used carefully, and the exposition is clear. If the main results are accepted, they provide a powerful new tool for studying singularities via characteristic classes, with the minimal exponent threshold behavior encoded directly in the spectral classes. The paper also gives clean new proofs of the isolated spectrum formulas, which are of independent interest.

major comments (2)
  1. [§4.4, Eq. (25)] The identification of the lowest non-zero graded de Rham piece with a pullback sheaf π^*(F) relies crucially on [10, Lemma 2.6], quoted from an unpublished preprint by one of the authors. This lemma is used in Theorem 4.7 and Corollary 4.10, and the converse directions of Theorems 4.21 and 4.24 both depend on the pullback form of this lowest piece. Since the lemma is not proven here, the main converses are not established independently. The authors should either supply a full proof of (25) within the paper or explicitly state it as an assumption and ensure the preprint is publicly available and verifiable for referees and readers.
  2. [§4.7, proof of Theorem 4.24] In the converse part of the proof, the authors assert that from td_*([π^*(F)]) ≠ 0 one obtains td_*([D_coh(π^*(F))]) ≠ 0 by citing [15, Example 18.3.19]. This is not a routine consequence of the cited example; duality under the Todd class transformation is subtle for arbitrary coherent sheaves on singular varieties. The authors should spell out the precise statement they are using and give a justification, or provide a direct argument showing that the Todd class of the Grothendieck dual is non-zero in this specific geometric setting. Without this, the converse for the lower-bound vanishing is incomplete.
minor comments (4)
  1. [§4.4, line before Eq. (24)] The phrase "monodromic module" should also mention "mixed Hodge module" for consistency with the rest of the section.
  2. [§4.6, proof of Theorem 4.21] In the long display after the sentence "Finally, this gives the vanishings", the case λ=1 lists p≤q but the preceding condition for λ=1 was p≤q; this is consistent, so no change needed, but the typography could be clarified.
  3. [References] Reference [10] is an arXiv preprint; the authors should indicate its current status (submitted, under review) or make the relevant lemma available in a published or easily accessible form.
  4. [Abstract] The abstract contains a typographical space error in "local complete intersection s"; this should be corrected.

Circularity Check

1 steps flagged · score 4.0 of 10

Converse threshold theorems are load-bearing on [10, Lemma 2.6], an unreviewed self-citation; no fitted-input or definitional circularity found.

  1. self citation load bearing [Section 4.4, equation (25), and proof of Theorem 4.21]
    "In fact, by [10, Lemma 2.6] that natural map is an isomorphism in this case. Thus, if Fσ M λ +Z = ⊕ℓ≥0 Fσ M r−1+λ +ℓ, we have an isomorphism (25) π ∗(Fσ M r−1+λ ) = (Fσ M r−1+λ )[z1, . . .,zr] ∼ = F σ M λ +Z. ... By the isomorphism (25), this is a coherent sheaf of the form π ∗(F ) for F some coherent sheaf on Σ X = Sing(X )."

    The converse of Theorem 4.21 (and similarly Theorem 4.24) needs the lowest non-vanishing graded de Rham piece of ϕ h,µ to be a pullback sheaf π*(F), because it then concludes s*td*(π*F) ≠ 0 from projectivity of Σ X. That pullback identification is not proved in this paper; it is exactly the content imported from [10, Lemma 2.6], an unpublished preprint by one of the present authors. The central converse claim is therefore delegated to a self-citation rather than derived here. This is load-bearing self-citation (pattern 3), not a definitional or fitted-parameter reduction; the forward vanishings and the independent b-function definition of ~α remain non-circular.

full rationale

The main construction is not circular in the definitional or fitted-parameter sense: the minimal exponent is defined in §4.1 from the Bernstein-Sato polynomial, the spectral Hirzebruch-Milnor classes are defined in §4.5 from the vanishing-cycle mixed Hodge module of the deformation to the normal cone, and no parameter is fitted to the class outputs. The forward vanishings in Theorems 4.21 and 4.24 follow from Theorem 4.7, which is derived from the peer-reviewed V-filtration characterization [7, Thm 4.3] (with [5] for the monodromic Hodge piece), and do not rely on [10]. The isolated-spectrum applications in Corollaries 4.23 and 4.25 are consistent with, and partly re-derive, results already in [10]; Remark 4.22 and the introduction acknowledge this. The only load-bearing reliance on an unverified self-citation is the use of [10, Lemma 2.6] to obtain the pullback form (25), which is then used in Corollary 4.10 and in the projective converse directions of Theorems 4.21 and 4.24. Because that lemma is from an unpublished preprint by one of the authors and no proof is supplied, the converse half of the central claim is not self-contained. This raises the circularity score to 4: there is some self-citation that is load-bearing, but the central vanishing results and the minimal-exponent input have independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim has no fitted constants or invented entities. It rests on the full Saito mixed Hodge module formalism, the V-filtration criterion of [7, Thm 4.3] as a black box, the technical hypotheses lct(X)>r-1 and projectivity of Sing(X) in the converse directions, and a gluing lemma quoted from the unpublished preprint [10]. These are listed as axioms because the paper does not prove them from first principles; none is tailored to force the target theorem in an ad hoc way.

assumptions (5)
  • standard math Saito's theory of algebraic mixed Hodge modules, including vanishing and nearby cycle functors, duality, and Tate twists, is used as established background.
    Invoked throughout Section 4, e.g., the triangle (19), Definition 4.11, and the duality isomorphism (54). No proofs are given, which is normal for this area but makes the claim depend on a large body of external theory.
  • domain assumption The V-filtration criterion of Chen-Dirks-Mustata-Olano [7, Thm 4.3] characterizes the minimal exponent ~α(X) by vanishing of filtered graded V-pieces of B_f.
    Theorem 4.7 and all subsequent minimal-exponent bounds derive from this prior theorem. It is stated, not proved, in this paper.
  • domain assumption The gluing lemma [10, Lemma 2.6] and the monodromic module isomorphisms (23)-(25) hold as stated.
    Used in the proofs of Theorem 4.7 and Corollary 4.10 to identify the lowest Hodge piece as a pullback along the vector bundle π. [10] is an unreviewed preprint by one of the authors.
  • domain assumption The technical hypotheses lct(X)>r-1 and projectivity of Sing(X) are satisfied in the statements where they appear.
    The formula (21) and the converse directions of Theorems 4.21, 4.24, 4.29, and 4.30 explicitly require these conditions; the claims are not asserted without them.
  • domain assumption X is a reduced local complete intersection subvariety of pure codimension r in a smooth complex algebraic variety Y, over C.
    This is the setup from the abstract and Section 4.1; all notations such as CXY, ~Y, and B_f depend on it.

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Pith. "Pith review of Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities." pith.science (2026). https://pith.science/paper/TW4ZX3MC

@misc{pith2026250206537,
  author       = {Pith},
  title        = {Pith review of: Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TW4ZX3MC}},
  note         = {Machine review of arXiv:2502.06537}
}
read the original abstract

In this paper we use the deformation to the normal cone and the corresponding Verdier-Saito specialization to define and study (spectral) Hirzebruch-Milnor type homology characteristic classes for local complete intersections. Our main results describe vanishing properties of these classes in relation to the minimal exponent. As applications, we show how Hirzebruch-Milnor classes of local complete intersections with a projective singular locus can be used to detect higher Du Bois and higher rational singularities.

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Works this paper leans on

39 extracted references · 34 canonical work pages

  1. [21]

    Maxim, M

    L. Maxim, M. Saito, J. Sch¨ urmann, Hirzebruch-Milnor classes of complete intersections Adv. Math. 241 (2013), 220–245

    work page 2013

  2. [10]

    Some applications of microlocalization for local complete intersection subvarieties

    B. Dirks, Some applications of microlocalization for local complete intersection subvarieties , arXiv:2310.15277

    work page Pith review arXiv

  3. [7]

    Q. Chen, B. Dirks, M. Mustat ¸˘ a, S. Olano, V -filtrations and minimal exponents for local complete inte r- sections, J. Reine Angew. Math. 811 (2024), 219–256

    work page 2024

  4. [1]

    P . Baum, W . Fulton, R. MacPherson, Riemann-Roch for singular varieties , Inst. Hautes ´Etudes Sci. Publ. Math. 45 (1975), 101–145

    work page 1975

  5. [2]

    Brasselet, J

    J.-P . Brasselet, J. Sch¨ urmann, S. Y okura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), no. 1, 1–55

    work page 2010

  6. [3]

    Budur, M

    N. Budur, M. Mustat ¸˘ a, M. Saito,Bernstein-Sato polynomials of arbitrary varieties , Compos. Math. 142 (2006), no. 3, 779–797

    work page 2006

  7. [4]

    S. E. Cappell, L. Maxim, J. Sch¨ urmann, J. L. Shaneson, Characteristic classes of complex hypersurfaces Adv. Math. 225 (2010), no. 5, 2616–2647. 34 BRADLEY DIRKS, LAURENT ¸ IU MAXIM, AND SEBASTI ´AN OLANO

    work page 2010

  8. [5]

    Q. Chen, B. Dirks, On V -filtration, Hodge filtration and F ourier transform , Selecta Math. (N.S.) 29 (2023), no. 4, Paper No. 50, 76 pp

    work page 2023

Show all 39 references
  1. [6]

    Q. Chen, B. Dirks, M. Mustat ¸˘ a,The minimal exponent and k-rationality for locally complete intersections, J. ´Ec. polytech. Math. 11 (2024), 849–873

    2024

  2. [8]

    Denef, F

    J. Denef, F. Loeser, Motivic exponential integrals and a motivic Thom-Sebastia ni theorem, Duke Math. J. 99 (1999), no. 2, 285–309

    1999

  3. [9]

    Dimca, P

    A. Dimca, P . Maisonobe, M. Saito, Spectrum and multiplier ideals of arbitrary subvarieties , Ann. Inst. Fourier 61 (2011), no. 4, 1633–1653

    2011

  4. [11]

    Dirks, S

    B. Dirks, S. Olano, D. Raychaudhury, A Hodge theoretic generalization of Q-homology manifolds , arXiv:2501.14065

    arXiv

  5. [12]

    Du Bois, Complexe de de Rham filtr ´e d’une vari ´et´e singuli `ere, Bull

    P . Du Bois, Complexe de de Rham filtr ´e d’une vari ´et´e singuli `ere, Bull. Soc. Math. France 109 (1981), no. 1, 41–81

    1981

  6. [13]

    Friedman, R

    R. Friedman, R. Laza, The higher Du Bois and higher rational properties for isolat ed singularities , J. Algebraic Geom. 33 (2024), no. 3, 493–520

    2024

  7. [14]

    Friedman, R

    R. Friedman, R. Laza, Higher Du Bois and higher rational singularities . Appendix by Morihiko Saito. Duke Math. J. 173 (2024), no. 10, 1839–1881

    2024

  8. [15]

    Fulton, Intersection Theory, Springer-V erlag, 1984

    W . Fulton, Intersection Theory, Springer-V erlag, 1984

    1984

  9. [16]

    Singulariti es and its applications

    T. Gaffney, J.J. Nu˜ no-Ballesteros, B. Or´ efice-Okamo to, M.A.S. Ruas, R. Wik-Atique, Com- plete intersections with isolated singularities (ICIS), A lgebraic methods and singulari- ties, lecture notes for 2022 CIMP A Research School “Singulariti es and its applications”, http...

    2022

  10. [17]

    Guibert, F

    G. Guibert, F. Loeser, M. Merle, Iterated vanishing cycles, convolution, and a motivic anal ogue of a conjecture of Steenbrink, Duke Math. J. 132 (2006), no. 3, 409–457

    2006

  11. [18]

    Ishii, On isolated Gorenstein singularities , Math

    S. Ishii, On isolated Gorenstein singularities , Math. Ann. 270 (1985), 541–554

    1985

  12. [19]

    Jung, I.-K

    S.-J. Jung, I.-K. Kim, M. Saito, Y . Y oon, Higher Du Bois singularities of hypersurfaces , Proc. Lond. Math. Soc. (3), 125 (2022), no. 3, 543–567

    2022

  13. [20]

    S. J. Kov´ acs,Rational, log canonical, Du Bois singularities: on the conj ectures of Koll ´ar and Steenbrink, Compos. Math. 118 (1999), no. 2, 123–133

    1999

  14. [22]

    Maxim, M

    L. Maxim, M. Saito, J. Sch¨ urmann, Spectral Hirzebruch-Milnor classes of singular hypersurf aces, Math. Ann. 377 (2020), no. 1-2, 281–315

    2020

  15. [23]

    Maxim, M

    L. Maxim, M. Saito, R. Y ang, Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singulariti es, Int. Math. Res. Not. IMRN 2024, no. 16, 11977–11988

    2024

  16. [24]

    Maxim, J

    L. Maxim, J. Sch¨ urmann, Characteristic classes of mixed Hodge modules and applicat ions, Schubert va- rieties, equivariant cohomology and characteristic class es—IMP ANGA 15, EMS Ser. Congr. Rep., pages 159–202. Eur. Math. Soc., Z¨ urich, 2018

    2018

  17. [25]

    Maxim, R

    L. Maxim, R. Y ang, Higher Du Bois and higher rational singularities of hypersu rfaces, arXiv:2301.09084

    arXiv

  18. [26]

    Mustat ¸˘ a, S

    M. Mustat ¸˘ a, S. Olano, M. Popa, J. Witaszek, The Du Bois complex of a hypersurface and the minimal exponent, Duke Math. J. 172 (2023), no. 7, 1411–1436

    2023

  19. [27]

    Mustat ¸˘ a, M

    M. Mustat ¸˘ a, M. Popa,On k-rational and k-Du Bois local complete intersections , arXiv:2207.08743

    arXiv

  20. [28]

    M. Popa, S. G. Park, Lefschetz theorems, Q-factoriality, and Hodge symmetry for singular varieties , arXiv:2410.15638

    arXiv

  21. [29]

    Saito, Modules de Hodge polarisables , Publ

    M. Saito, Modules de Hodge polarisables , Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, pp. 849–995

    1988

  22. [30]

    Saito, Mixed Hodge Modules , Publ

    M. Saito, Mixed Hodge Modules , Publ. RIMS, Kyoto Univ. 26 (1990), 221–333. CHARACTERISTIC CLASSES AND HIGHER SINGULARITIES 35

    1990

  23. [31]

    Saito, On Steenbrink’s conjecture , Math

    M. Saito, On Steenbrink’s conjecture , Math. Ann. 289 (1991), 703–716

    1991

  24. [32]

    Saito, On microlocal b-function, Bull

    M. Saito, On microlocal b-function, Bull. Soc. Math. France 122 (1994), no. 2, 163–184

    1994

  25. [33]

    Saito, Examples of Hirzebruch-Milnor classes of projective hyper surfaces detecting higher du Bois or rational singularities, arXiv:2303.04724

    M. Saito, Examples of Hirzebruch-Milnor classes of projective hyper surfaces detecting higher du Bois or rational singularities, arXiv:2303.04724

    arXiv

  26. [34]

    Sch¨ urmann, A generalized V erdier-type Riemann-Roch theorem for Chern -Schwartz-MacPherson classes, arXiv:math/0202175

    J. Sch¨ urmann, A generalized V erdier-type Riemann-Roch theorem for Chern -Schwartz-MacPherson classes, arXiv:math/0202175

    arXiv

  27. [35]

    Sch¨ urmann, T opology of singular spaces and constructible sheaves , Monografie Matematyczne 63, Birkh¨ auser V erlag, Basel, 2003

    J. Sch¨ urmann, T opology of singular spaces and constructible sheaves , Monografie Matematyczne 63, Birkh¨ auser V erlag, Basel, 2003

    2003

  28. [36]

    Sch¨ urmann,Specialization of motivic Hodge-Chern classes , arXiv:0909.3478

    J. Sch¨ urmann,Specialization of motivic Hodge-Chern classes , arXiv:0909.3478

    arXiv

  29. [37]

    Sch¨ urmann, Characteristic classes of mixed Hodge modules , Topology of stratified spaces, 419–470, Math

    J. Sch¨ urmann, Characteristic classes of mixed Hodge modules , Topology of stratified spaces, 419–470, Math. Sci. Res. Inst. Publ., 58, Cambridge Univ. Press, Camb ridge, 2011

    2011

  30. [38]

    V erdier, Le th ´eor`eme de Riemann-Roch pour les intersections compl `etes, Ast´ erisque 36-37 (1976), 189–228

    J.-L. V erdier, Le th ´eor`eme de Riemann-Roch pour les intersections compl `etes, Ast´ erisque 36-37 (1976), 189–228

    1976

  31. [39]

    V erdier, Sp´ecialisation de faisceaux et monodromie mod ´er´ee, Analyse et Topologie sur les Espaces singuliers (II-III), Ast´ erisque 101–102 (1983), 332–364

    J.-L. V erdier, Sp´ecialisation de faisceaux et monodromie mod ´er´ee, Analyse et Topologie sur les Espaces singuliers (II-III), Ast´ erisque 101–102 (1983), 332–364 . SCHOOL OF MATHEMATICS , 1 E INSTEIN DRIVE , P RINCETON , N EW JERSEY 08540, USA Email address: bradley.dirks@...

    1983

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