logo AFST
Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 689-771.

We construct complete Kähler metrics on the nonsingular set of a subvariety X of a compact Kähler manifold. To that end, we develop (i) a constructive method for replacing a sequence of blow-ups along smooth centers, with a single blow-up along a product of coherent ideals corresponding to the centers and (ii) an explicit local formula for a Chern form associated to this ‘singular’ blow-up. Our metrics have a particularly simple local formula of a sum of the original metric and of the pull back of the classical Poincaré metric on the punctured disc by a ‘size-function’ S I of a coherent ideal I used to resolve the singularities of X by a ‘singular’ blow-up, where (S I ) 2 := j=1 r f j 2 and the f j ’s are the local generators of the ideal I . Our proof of (i) makes use of our generalization of Chow’s theorem for coherent ideals. We prove Saper type growth for our metric near the singular set and local boundedness of the gradient of a local generating function for our metric, motivated by results of Donnelly-Fefferman, Ohsawa, and Gromov on the vanishing of certain L 2 -cohomology groups.

Nous construisons des métriques complètes Kähleriennes sur le lieu non-singulier d’une sous-variété X d’une variété compacte Kählerienne lisse. A cet effet, nous développons : (i) une méthode constructive pour le remplacement d’une suite d’éclatements le long des centres lisses par un seul éclatement le long d’un produit d’idéaux cohérents et (ii) une formule locale explicite pour une forme de Chern associée à cet éclatement. Nos métriques sont décrites par une formule locale particulièrement simple comme la somme de la métrique de départ et le tire-en-arrière de la métrique de Poincaré classique sur le disque épointé par une ‘fonction de grandeur’ S I de l’idéal cohérent I utilisé pour la résolution des singularités de X, ou (S I ) 2 := j=1 r f j 2 et les f j sont des générateurs locaux de I. Notre preuve de (i) utilise notre généralisation du théorème de Chow pour les idéaux cohérents. Nous montrons que la vitesse de croissance de notre métrique près du lieu singulier est de type Saper ainsi que le fait que le gradient d’une fonction génératrice locale de notre métrique est borné. Cela est motivé par les résultats de Donnelly-Fefferman, Ohsawa, et Gromov sur l’annulation de certains groupes de cohomologie L 2 .

Received:
Accepted:
DOI: 10.5802/afst.1134
Caroline Grant Melles 1; Pierre Milman 2

1 Mathematics Department, United States Naval Academy, 572C Holloway Rd, Annapolis, Maryland 21402-5002, United States of America.
2 Department of Mathematics, University of Toronto, 40 St George St, Toronto, Ontario M5S 2E4, Canada.
@article{AFST_2006_6_15_4_689_0,
     author = {Caroline Grant Melles and Pierre Milman},
     title = {Classical {Poincar\'e} metric pulled back off singularities using a {Chow-type} theorem and desingularization},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {689--771},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 15},
     number = {4},
     year = {2006},
     doi = {10.5802/afst.1134},
     mrnumber = {2295209},
     zbl = {1207.32016},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1134/}
}
TY  - JOUR
TI  - Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2006
DA  - 2006///
SP  - 689
EP  - 771
VL  - Ser. 6, 15
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1134/
UR  - https://www.ams.org/mathscinet-getitem?mr=2295209
UR  - https://zbmath.org/?q=an%3A1207.32016
UR  - https://doi.org/10.5802/afst.1134
DO  - 10.5802/afst.1134
LA  - en
ID  - AFST_2006_6_15_4_689_0
ER  - 
%0 Journal Article
%T Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2006
%P 689-771
%V Ser. 6, 15
%N 4
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1134
%R 10.5802/afst.1134
%G en
%F AFST_2006_6_15_4_689_0
Caroline Grant Melles; Pierre Milman. Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 689-771. doi : 10.5802/afst.1134. https://afst.centre-mersenne.org/articles/10.5802/afst.1134/

[BM1] Bierstone (E.); Milman (P.) Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997), pp. 207-302 | MR: 1440306 | Zbl: 0896.14006

[BM2] Bierstone (E.); Milman (P.D.) Desingularization of Toric and Binomial Varieties, J. Algebraic Geom., Volume 15 (2006), pp. 443-486 | MR: 2219845 | Zbl: 1120.14009

[C] Cheeger (J.) On the Hodge Theory of Riemannian Pseudomanifolds, Proc. Symp. Pure Math., American Math. Soc., Volume 36 (1980), pp. 91-146 | MR: 573430 | Zbl: 0461.58002

[CGM] Cheeger (J.); Goresky (M.); MacPherson (R.) L 2 -Cohomology and Intersection Homology of Singular Algebraic Varieties, Seminar on Differential Geometry (Annals of Mathematics Studies) (1982) no. 102, pp. 303-340 | MR: 645745 | Zbl: 0503.14008

[DF] Donnelly (H.); Fefferman (C.) L 2 -cohomology and index theorem for the Bergman metric, Ann. Math., Volume 118 (1983), pp. 593-618 | MR: 727705 | Zbl: 0532.58027

[F] Fischer (G.) Complex Analytic Geometry, Lecture Notes in Math., Springer-Verlag, Berlin Heidelberg, 1976 no. 538 | MR: 430286 | Zbl: 0343.32002

[Ful] Fulton (W.) Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd 2, Springer-Verlag, Berlin Heidelberg, 1984 | MR: 732620 | Zbl: 0885.14002

[GH] Griffiths (P.); Harris (J.) Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978 | MR: 507725 | Zbl: 0408.14001

[GM1] Milman (P.) Grant (C.) Metrics for Singular Analytic Spaces, Pac. J. Math., Volume 168 (1995), pp. 61-156 | MR: 1331995 | Zbl: 0822.32004

[GM3] Grant Melles (C.); Milman (P.) Explicit Construction of Complete Kähler Metrics of Saper Type by Desingularization (1999) (Preprint math.AG/9907056, p. 1–43) | MR: 1792150

[GM2] Grant Melles (C.); Milman (P.) Single-Step Combinatorial Resolution via Coherent Sheaves of Ideals, Singularities in Algebraic and Analytic Geometry (Contemporary Mathematics) (2000) no. 266, pp. 77-88 | MR: 1792150 | Zbl: 0973.14007

[GoM] Goresky (M.); MacPherson (R.) Intersection Homology II, Invent. Math., Volume 71 (1983), pp. 77-129 | MR: 696691 | Zbl: 0529.55007

[Gro] Gromov (M.) Kähler hyperbolicity and L 2 -Hodge theory, J. Diff. Geom., Volume 33 (1991), pp. 263-292 | MR: 1085144 | Zbl: 0719.53042

[GrR2] Grauert (H.); Remmert (R.) Theory of Stein Spaces, Grundlehren der mathematischen Wissenschaften, Volume 236, Springer-Verlag, New York, 1979 | MR: 580152 | Zbl: 0433.32007

[GrR1] Grauert (H.); Remmert (R.) Coherent Analytic Sheaves, Grundlehren der mathematischen Wissenschaften, Volume 265, Springer-Verlag, Berlin Heidelberg,, 1984 | MR: 755331 | Zbl: 0537.32001

[GuR] Gunning (R.); Rossi (H.) Analytic Functions of Several Complex Variables, Prentice-Hall Inc., Englewood Cliffs, NJ, 1965 | MR: 180696 | Zbl: 0141.08601

[Ha2] Hartshorne (R.) Ample Subvarieties of Algebraic Varieties (Lecture Notes in Math.) Volume 156, Springer-Verlag, Heidelberg, 1970 | MR: 282977 | Zbl: 0208.48901

[Ha1] Hartshorne (R.) Graduate Texts in Mathematics, Algebraic Geometry, Springer-Verlag, New York, 1977 no. 52 | MR: 463157 | Zbl: 0367.14001

[Hi] Hironaka (H.) Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. Math., Volume 79 (1964), pp. 109-326 | MR: 199184 | Zbl: 0122.38603

[Ho] Hörmander (L.) An Introduction to Complex Analysis in Several Variables, North-Holland, New York, 1973 | MR: 344507 | Zbl: 0138.06203

[HR] Rossi (H.) Hironaka (H.) On the Equivalence of Imbeddings of Exceptional Complex Spaces, Math. Annalen, Volume 156 (1964), pp. 313-333 | MR: 171784 | Zbl: 0136.20801

[I] Iitaka (S.) Graduate Texts in Mathematics, Algebraic Geometry, Springer-Verlag, New York, 1982 no. 76 | MR: 637060 | Zbl: 0491.14006

[K] Kunz (E.) Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985 | MR: 789602 | Zbl: 0563.13001

[Lo] Lojasiewicz (S.) Introduction to Complex Analytic Geometry, Birkhauser, Basel, 1991 | MR: 1131081 | Zbl: 0747.32001

[LT] Teissier (B.) Lejeune-Jalabert (M.) 1, Clôture integrale des ideaux et equisingularité (1974)

[M] Mumford (D.) Algebraic Geometry I Complex Projective Varieties, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin Heidelberg, 1976 no. 221 | MR: 266911 | Zbl: 0356.14002

[Ma] Matsumura (H.) Commutative Algebra, W. A. Benjamin Co., New York, 1970 | MR: 453732 | Zbl: 0356.14002

[O] Ohsawa (T.) Hodge Spectral Sequence on Compact Kähler Spaces, Publ. R.I.M.S., Kyoto Univ., Volume 23 (1987), pp. 265-274 | MR: 890919 | Zbl: 0626.32029

[Sa1] Saper (L.) L 2 -cohomology and intersection homology of certain algebraic varieties with isolated singularities, Invent. Math., Volume 82 (1985), pp. 207-255 | MR: 809713 | Zbl: 0611.14018

[Sa2] Saper (L.) L 2 -cohomology of Kähler varieties with isolated singularities, J. Diff. Geom., Volume 36 (1992), pp. 89-161 | MR: 1168983 | Zbl: 0780.14010

[Sh] Shafarevich (I.) Basic Algebraic Geometry Volume 2, Springer-Verlag, Berlin Heidelberg, 1994 | Zbl: 0797.14001

[Sp] Spivakovsky (M.) Valuations in Function Fields of Surfaces, Am. J. Math., Volume 112 (1990), pp. 107-156 | MR: 1037606 | Zbl: 0716.13003

[W] Wells (R.O.) Graduate Texts in Mathematics, Differential Analysis on Complex Manifolds, Springer-Verlag, New York, 1980 no. 65 | MR: 608414 | Zbl: 0435.32004

[ZS] Zariski (O.); Samuel (P.) Graduate Texts in Mathematics, Commutative Algebra Volume II, Springer-Verlag, New York, 1960 no. 29 | MR: 120249 | Zbl: 0322.13001

[Zu1] Zucker (S.) Hodge theory with degenerating coefficients: L 2 cohomology in the Poincaré metric, Ann. Math., Volume 109 (1979), pp. 415-476 | MR: 534758 | Zbl: 0446.14002

[Zu2] Zucker (S.) L 2 cohomology of Warped Products and Arithmetic Groups, Invent. Math., Volume 70 (1982), pp. 169-218 | MR: 684171 | Zbl: 0508.20020

Cited by Sources: