Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization

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.

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 4, page 689-771
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Grant Melles, Caroline, and Milman, Pierre. "Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization." Annales de la faculté des sciences de Toulouse Mathématiques 15.4 (2006): 689-771. <http://eudml.org/doc/10020>.

@article{GrantMelles2006,
abstract = {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 $(S_I)^2 := \sum _\{j=1\}^r \{\mid f_j \mid \}^2$ 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.},
affiliation = {Mathematics Department, United States Naval Academy, 572C Holloway Rd, Annapolis, Maryland 21402-5002, United States of America.; Department of Mathematics, University of Toronto, 40 St George St, Toronto, Ontario M5S 2E4, Canada.},
author = {Grant Melles, Caroline, Milman, Pierre},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {4},
pages = {689-771},
publisher = {Université Paul Sabatier, Toulouse},
title = {Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization},
url = {http://eudml.org/doc/10020},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Grant Melles, Caroline
AU - Milman, Pierre
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
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 4
SP - 689
EP - 771
AB - 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 $(S_I)^2 := \sum _{j=1}^r {\mid f_j \mid }^2$ 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.
LA - eng
UR - http://eudml.org/doc/10020
ER -

References

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  1. Bierstone (E.), Milman (P.), Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207-302 Zbl0896.14006MR1440306
  2. Bierstone (E.), Milman (P.D.), Desingularization of Toric and Binomial Varieties, J. Algebraic Geom. 15 (2006), 443-486 Zbl1120.14009MR2219845
  3. Cheeger (J.), On the Hodge Theory of Riemannian Pseudomanifolds, Proc. Symp. Pure Math., American Math. Soc. 36 (1980), 91-146 Zbl0461.58002MR573430
  4. Cheeger (J.), Goresky (M.), MacPherson (R.), L 2 -Cohomology and Intersection Homology of Singular Algebraic Varieties, Seminar on Differential Geometry (1982), 303-340, Princeton University Press, Princeton, NJ Zbl0503.14008MR645745
  5. Donnelly (H.), Fefferman (C.), L 2 -cohomology and index theorem for the Bergman metric, Ann. Math. 118 (1983), 593-618 Zbl0532.58027MR727705
  6. Fischer (G.), Complex Analytic Geometry, (1976), Springer-Verlag, Berlin Heidelberg Zbl0343.32002MR430286
  7. Fulton (W.), Intersection Theory, (1984), Springer-Verlag, Berlin Heidelberg Zbl0885.14002MR732620
  8. Griffiths (P.), Harris (J.), Principles of Algebraic Geometry, (1978), Wiley-Interscience, New York Zbl0408.14001MR507725
  9. Milman (P.) Grant (C.), Metrics for Singular Analytic Spaces, Pac. J. Math. 168 (1995), 61-156 Zbl0822.32004MR1331995
  10. Grant Melles (C.), Milman (P.), Single-Step Combinatorial Resolution via Coherent Sheaves of Ideals, Singularities in Algebraic and Analytic Geometry (2000), 77-88, American Mathematical Society, Providence, RI Zbl0973.14007MR1792150
  11. Grant Melles (C.), Milman (P.), Explicit Construction of Complete Kähler Metrics of Saper Type by Desingularization, (1999) MR1792150
  12. Goresky (M.), MacPherson (R.), Intersection Homology II, Invent. Math. 71 (1983), 77-129 Zbl0529.55007MR696691
  13. Gromov (M.), Kähler hyperbolicity and L 2 -Hodge theory, J. Diff. Geom. 33 (1991), 263-292 Zbl0719.53042MR1085144
  14. Grauert (H.), Remmert (R.), Coherent Analytic Sheaves, 265 (1984), Springer-Verlag, Berlin Heidelberg, Zbl0537.32001MR755331
  15. Grauert (H.), Remmert (R.), Theory of Stein Spaces, 236 (1979), Springer-Verlag, New York Zbl0433.32007MR580152
  16. Gunning (R.), Rossi (H.), Analytic Functions of Several Complex Variables, (1965), Prentice-Hall Inc., Englewood Cliffs, NJ Zbl0141.08601MR180696
  17. Hartshorne (R.), Graduate Texts in Mathematics, Algebraic Geometry (1977), Springer-Verlag, New York Zbl0367.14001MR463157
  18. Hartshorne (R.), Ample Subvarieties of Algebraic Varieties, 156 (1970), Springer-Verlag, Heidelberg Zbl0208.48901MR282977
  19. Hironaka (H.), Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. Math. 79 (1964), 109-326 Zbl0122.38603MR199184
  20. Hörmander (L.), An Introduction to Complex Analysis in Several Variables, (1973), North-Holland, New York Zbl0138.06203MR344507
  21. Rossi (H.) Hironaka (H.), On the Equivalence of Imbeddings of Exceptional Complex Spaces, Math. Annalen 156 (1964), 313-333 Zbl0136.20801MR171784
  22. Iitaka (S.), Graduate Texts in Mathematics, Algebraic Geometry (1982), Springer-Verlag, New York Zbl0491.14006MR637060
  23. Kunz (E.), Introduction to Commutative Algebra and Algebraic Geometry, (1985), Birkhäuser, Boston Zbl0563.13001MR789602
  24. Lojasiewicz (S.), Introduction to Complex Analytic Geometry, (1991), Birkhauser, Basel Zbl0747.32001MR1131081
  25. Teissier (B.) Lejeune-Jalabert (M.), Clôture integrale des ideaux et equisingularité, (1974), Publ. Inst. Fourier 
  26. Mumford (D.), Algebraic Geometry I Complex Projective Varieties, (1976), Springer-Verlag, Berlin Heidelberg Zbl0356.14002MR266911
  27. Matsumura (H.), Commutative Algebra, (1970), W. A. Benjamin Co., New York Zbl0356.14002MR453732
  28. Ohsawa (T.), Hodge Spectral Sequence on Compact Kähler Spaces, Publ. R.I.M.S., Kyoto Univ. 23 (1987), 265-274 Zbl0626.32029MR890919
  29. Saper (L.), L 2 -cohomology and intersection homology of certain algebraic varieties with isolated singularities, Invent. Math. 82 (1985), 207-255 Zbl0611.14018MR809713
  30. Saper (L.), L 2 -cohomology of Kähler varieties with isolated singularities, J. Diff. Geom. 36 (1992), 89-161 Zbl0780.14010MR1168983
  31. Shafarevich (I.), Basic Algebraic Geometry, 2 (1994), Springer-Verlag, Berlin Heidelberg Zbl0797.14001
  32. Spivakovsky (M.), Valuations in Function Fields of Surfaces, Am. J. Math. 112 (1990), 107-156 Zbl0716.13003MR1037606
  33. Wells (R.O.), Graduate Texts in Mathematics, Differential Analysis on Complex Manifolds (1980), Springer-Verlag, New York Zbl0435.32004MR608414
  34. Zariski (O.), Samuel (P.), Graduate Texts in Mathematics, Commutative Algebra Volume II (1960), Springer-Verlag, New York Zbl0322.13001MR120249
  35. Zucker (S.), Hodge theory with degenerating coefficients: L 2 cohomology in the Poincaré metric, Ann. Math. 109 (1979), 415-476 Zbl0446.14002MR534758
  36. Zucker (S.), L 2 cohomology of Warped Products and Arithmetic Groups, Invent. Math. 70 (1982), 169-218 Zbl0508.20020MR684171

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