Volume 23, issue 7 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 7, 2855–3306
Issue 6, 2405–2853
Issue 5, 1907–2404
Issue 4, 1435–1905
Issue 3, 937–1434
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Resolution of singularities and geometric proofs of the Łojasiewicz inequalities

Paul M N Feehan

Geometry & Topology 23 (2019) 3273–3313
Bibliography
1 O M Abderrahmane, On the Łojasiewicz exponent and Newton polyhedron, Kodai Math. J. 28 (2005) 106 MR2122194
2 D Adams, L Simon, Rates of asymptotic convergence near isolated singularities of geometric extrema, Indiana Univ. Math. J. 37 (1988) 225 MR963501
3 J M Aroca, H Hironaka, J L Vicente, The theory of the maximal contact, 29, Consejo Superior de Investigaciones Cientificas (1975) 135 MR0444999
4 J M Aroca, H Hironaka, J L Vicente, Desingularization theorems, 30, Consejo Superior de Investigaciones Científicas (1977) MR480502
5 M F Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970) 145 MR256156
6 D M Austin, P J Braam, Morse–Bott theory and equivariant cohomology, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 123 MR1362827
7 A Banyaga, D E Hurtubise, A proof of the Morse–Bott lemma, Expo. Math. 22 (2004) 365 MR2075744
8 I N Bernstein, S I Gelfand, Meromorphy of the function Pλ, Funkcional. Anal. i Priložen. 3 (1969) 84 MR0247457
9 E Bierstone, P D Milman, Resolution of singularities, from: "Several complex variables" (editors M Schneider, Y T Siu), Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press (1999) 43 MR1748600
10 E Bierstone, P D Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5 MR972342
11 E Bierstone, P D Milman, Uniformization of analytic spaces, J. Amer. Math. Soc. 2 (1989) 801 MR1001853
12 E Bierstone, P D Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997) 207 MR1440306
13 C Bivià-Ausina, Multiplicity and Łojasiewicz exponent of generic linear sections of monomial ideals, Bull. Aust. Math. Soc. 91 (2015) 191 MR3314137
14 C Bivià-Ausina, S Encinas, Łojasiewicz exponents and resolution of singularities, Arch. Math. (Basel) 93 (2009) 225 MR2540788
15 C Bivià-Ausina, S Encinas, The Łojasiewicz exponent of a set of weighted homogeneous ideals, J. Pure Appl. Algebra 215 (2011) 578 MR2738373
16 C Bivià-Ausina, S Encinas, Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations, Rev. Mat. Complut. 26 (2013) 773 MR3068619
17 C Bivià-Ausina, T Fukui, Mixed Łojasiewicz exponents and log canonical thresholds of ideals, J. Pure Appl. Algebra 220 (2016) 223 MR3393458
18 R Bott, Nondegenerate critical manifolds, Ann. of Math. 60 (1954) 248 MR64399
19 R Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959) 313 MR110104
20 S Brzostowski, The Łojasiewicz exponent of semiquasihomogeneous singularities, Bull. Lond. Math. Soc. 47 (2015) 848 MR3403966
21 S Brzostowski, T Krasiński, G Oleksik, A conjecture on the Łojasiewicz exponent, J. Singul. 6 (2012) 124 MR2971310
22 N T N Bùi, T S Phạm, Computation of the Łojasiewicz exponent of nonnegative and nondegenerate analytic functions, Internat. J. Math. 25 (2014) MR3275820
23 R Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 (2003) 572 MR1986700
24 R Chill, A Haraux, M A Jendoubi, Applications of the Łojasiewicz–Simon gradient inequality to gradient-like evolution equations, Anal. Appl. (Singap.) 7 (2009) 351 MR2572850
25 T H Colding, W P Minicozzi II, Łojasiewicz inequalities and applications, from: "Surveys in differential geometry 2014 : Regularity and evolution of nonlinear equations" (editors H D Cao, R Schoen, S T Yau), Surv. Differ. Geom. 19, International (2015) 63 MR3381496
26 T H Colding, W P Minicozzi II, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. 182 (2015) 221 MR3374960
27 T H Colding, W P Minicozzi II, E K Pedersen, Mean curvature flow, Bull. Amer. Math. Soc. 52 (2015) 297 MR3312634
28 T C Collins, A Greenleaf, M Pramanik, A multi-dimensional resolution of singularities with applications to analysis, Amer. J. Math. 135 (2013) 1179 MR3117305
29 S D Cutkosky, Resolution of singularities, 63, Amer. Math. Soc. (2004) MR2058431
30 D D’Acunto, K Kurdyka, Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials, Ann. Polon. Math. 87 (2005) 51 MR2208535
31 C De Lellis, Łojasiewicz–Simon inequality, online resource (2013)
32 S Encinas, O Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998) 109 MR1654779
33 E Faber, H Hauser, Today’s menu : geometry and resolution of singular algebraic surfaces, Bull. Amer. Math. Soc. 47 (2010) 373 MR2651084
34 P M N Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, preprint (2016) arXiv:1409.1525v4
35 P M N Feehan, Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions, preprint (2017) arXiv:1706.09349
36 P M N Feehan, On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half, preprint (2018) arXiv:1803.11319
37 P M N Feehan, M Maridakis, Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functionals, preprint (2017) arXiv:1510.03815v4
38 P M N Feehan, M Maridakis, Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces, J. Reine Angew. Math. (2019)
39 T Fukui, Łojasiewicz type inequalities and Newton diagrams, Proc. Amer. Math. Soc. 112 (1991) 1169 MR1065945
40 A M Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen. 2 (1968) 18 MR0245831
41 A Gabrièlov, Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality, Selecta Math. 1 (1995) 113 MR1327229
42 H Grauert, R Remmert, Analytische Stellenalgebren, 176, Springer (1971) MR0316742
43 H Grauert, R Remmert, Coherent analytic sheaves, 265, Springer (1984) MR755331
44 M Greenblatt, An elementary coordinate-dependent local resolution of singularities and applications, J. Funct. Anal. 255 (2008) 1957 MR2462583
45 P Griffiths, J Harris, Principles of algebraic geometry, Wiley (1978) MR507725
46 F Guaraldo, P Macrì, A Tancredi, Topics on real analytic spaces, Friedr. (1986) MR1013362
47 J Gwoździewicz, The Łojasiewicz exponent of an analytic function at an isolated zero, Comment. Math. Helv. 74 (1999) 364 MR1710702
48 A Haraux, Positively homogeneous functions and the Łojasiewicz gradient inequality, Ann. Polon. Math. 87 (2005) 165 MR2208543
49 A Haraux, M A Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 (2007) 449 MR2328934
50 A Haraux, M A Jendoubi, The convergence problem for dissipative autonomous systems: classical methods and recent advances, Springer (2015) MR3380967
51 A Haraux, M A Jendoubi, O Kavian, Rate of decay to equilibrium in some semilinear parabolic equations, J. Evol. Equ. 3 (2003) 463 MR2019030
52 A Haraux, T S Phạm, On the gradient of quasi-homogeneous polynomials, Univ. Iagel. Acta Math. 49 (2011) 45 MR3076842
53 A Haraux, T S Phạm, On the Łojasiewicz exponents of quasi-homogeneous functions, J. Singul. 11 (2015) 52 MR3341438
54 R M Hardt, Stratification of real analytic mappings and images, Invent. Math. 28 (1975) 193 MR372237
55 R M Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1976) 207 MR454051
56 G H Hardy, J E Littlewood, G Pólya, Inequalities, Cambridge Univ. Press (1952) MR0046395
57 P Hartman, Ordinary differential equations, 38, SIAM (2002) MR1929104
58 R Hartshorne, Algebraic geometry, 52, Springer (1977) MR0463157
59 H Hauser, The Hironaka theorem on resolution of singularities (or : A proof we always wanted to understand), Bull. Amer. Math. Soc. 40 (2003) 323 MR1978567
60 H Hironaka, On resolution of singularities (characteristic zero), from: "Proc. Internat. Congr. Mathematicians" (editor V Stenström), Inst. Mittag-Leffler (1963) 507 MR0175898
61 H Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. 79 (1964) 109 MR0199184
62 H Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, II, Ann. of Math. 79 (1964) 205 MR0199184
63 H Hironaka, Introduction to real-analytic sets and real-analytic maps, Istituto Matematico “L Tonelli” dell’Università di Pisa (1973) MR0477121
64 H Hironaka, Subanalytic sets, from: "Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki" (editors Y Kusunoki, H Toda, M Yamaguti, H Yoshizawa), Kinokuniya (1973) 453 MR0377101
65 H Hironaka, Introduction to the theory of infinitely near singular points, 28, Consejo Superior de Investigaciones Científicas (1974) MR0399505
66 H Hironaka, Stratification and flatness, from: "Real and complex singularities", Sijthoff and Noordhoff (1977) 199 MR0499286
67 T Holm, Y Karshon, The Morse–Bott–Kirwan condition is local, Res. Math. Sci. 3 (2016) MR3579296
68 L Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958) 555 MR124734
69 S Z Huang, Gradient inequalities: with applications to asymptotic behavior and stability of gradient-like systems, 126, Amer. Math. Soc. (2006) MR2226672
70 S Ji, J Kollár, B Shiffman, A global Łojasiewicz inequality for algebraic varieties, Trans. Amer. Math. Soc. 329 (1992) 813 MR1046016
71 M Kashiwara, P Schapira, Sheaves on manifolds, 292, Springer (1994) MR1299726
72 F C Kirwan, Cohomology of quotients in symplectic and algebraic geometry, 31, Princeton Univ. Press (1984) MR766741
73 J Kollár, An effective Łojasiewicz inequality for real polynomials, Period. Math. Hungar. 38 (1999) 213 MR1756239
74 J Kollár, Lectures on resolution of singularities, 166, Princeton Univ. Press (2007) MR2289519
75 H J Kowalsky, Topologische Räume, 26, Birkhäuser (1961) 271 MR0121768
76 T Krasiński, G Oleksik, A Płoski, The Łojasiewicz exponent of an isolated weighted homogeneous surface singularity, Proc. Amer. Math. Soc. 137 (2009) 3387 MR2515408
77 N H Kuiper, C1–equivalence of functions near isolated critical points, from: "Symposium on infinite-dimensional topology" (editor R D Anderson), Ann. of Math. Studies 69, Princeton Univ. Press (1972) 199 MR0413161
78 H H Kuo, The Morse–Palais lemma on Banach spaces, Bull. Amer. Math. Soc. 80 (1974) 363 MR334274
79 K Kurdyka, T Mostowski, A Parusiński, Proof of the gradient conjecture of R Thom, Ann. of Math. 152 (2000) 763 MR1815701
80 K Kurdyka, A Parusiński, wf–stratification of subanalytic functions and the Łojasiewicz inequality, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994) 129 MR1260324
81 S Lang, Real and functional analysis, 142, Springer (1993) MR1216137
82 R Lazarsfeld, Positivity in algebraic geometry, I : Classical setting — line bundles and linear series, 48, Springer (2004) MR2095471
83 A Lenarcik, On the Łojasiewicz exponent of the gradient of a holomorphic function, from: "Singularities symposium : Łojasiewicz 70" (editors B Jakubczyk, W Pawłucki, J Stasica), Banach Center Publ. 44, Polish Acad. Sci. Inst. Math. (1998) 149 MR1677363
84 A Lenarcik, On the Łojasiewicz exponent of the gradient of a polynomial function, Ann. Polon. Math. 71 (1999) 211 MR1704300
85 E Lerman, Gradient flow of the norm squared of a moment map, Enseign. Math. 51 (2005) 117 MR2154623
86 B Lichtin, Estimation of Łojasiewicz exponents and Newton polygons, Invent. Math. 64 (1981) 417 MR632982
87 J M Lion, Inégalité de Łojasiewicz en géométrie pfaffienne, Illinois J. Math. 44 (2000) 889 MR1804312
88 S Łojasiewicz, Sur le problème de la division, Studia Math. 18 (1959) 87 MR107168
89 S Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, from: "Les équations aux dérivées partielles", Éditions du Centre National de la Recherche Scientifique (1963) 87 MR0160856
90 S Łojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 449 MR173265
91 S Łojasiewicz, Ensembles semi-analytiques, course notes (1965)
92 S Łojasiewicz, Sur les ensembles semi-analytiques, from: "Actes du Congrès International des Mathématiciens" (editors M Berger, J Dieudonné, J Leray, J L Lions, P Malliavin, J P Serre), Gauthiers-Villars (1971) 237 MR0425152
93 S Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, from: "Geometry seminars, 1982–1983", Univ. Stud. Bologna (1984) 115 MR771152
94 S Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble) 43 (1993) 1575 MR1275210
95 S Łojasiewicz, M A Zurro, On the gradient inequality, Bull. Polish Acad. Sci. Math. 47 (1999) 143 MR1686676
96 R Narasimhan, Introduction to the theory of analytic spaces, 25, Springer (1966) MR0217337
97 L Nicolaescu, An invitation to Morse theory, Springer (2011) MR2883440
98 M Oka, Łojasiewicz exponents of non-degenerate holomorohic and mixed functions, Kodai Math. J. 41 (2018) 620 MR3870707
99 G Oleksik, The Łojasiewicz exponent of nondegenerate singularities, Univ. Iagel. Acta Math. 47 (2009) 301 MR2743702
100 J Palis Jr., W de Melo, Geometric theory of dynamical systems: an introduction, Springer (1982) MR669541
101 A Płoski, Semicontinuity of the Łojasiewicz exponent, Univ. Iagel. Acta Math. 48 (2010) 103 MR2882792
102 J Råde, On the Yang–Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992) 123 MR1179335
103 J J Risler, D Trotman, Bi-Lipschitz invariance of the multiplicity, Bull. London Math. Soc. 29 (1997) 200 MR1425999
104 T Rodak, S Spodzieja, Effective formulas for the local Łojasiewicz exponent, Math. Z. 268 (2011) 37 MR2805423
105 E H Rothe, Morse theory in Hilbert space, Rocky Mountain J. Math. 3 (1973) 251 MR334275
106 L Schwartz, Division par une fonction holomorphe sur une variété analytique complexe, Summa Brasil. Math. 3 (1955) 181 MR139937
107 L Schwartz, Théorie des distributions à valeurs vectorielles, I, Ann. Inst. Fourier, Grenoble 7 (1957) 1 MR0107812
108 I R Shafarevich, Basic algebraic geometry, I: Varieties in projective space, Springer (2013) MR3100243
109 I R Shafarevich, Basic algebraic geometry, II: Schemes and complex manifolds, Springer (2013) MR3100288
110 L Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983) 525 MR727703
111 L Simon, Isolated singularities of extrema of geometric variational problems, from: "Harmonic mappings and minimal immersions" (editor E Giusti), Lecture Notes in Math. 1161, Springer (1985) 206 MR821971
112 L Simon, Theorems on regularity and singularity of energy minimizing maps, Birkhäuser (1996) MR1399562
113 K Smith, Resolution of singularities, lecture Notes (2016)
114 K E Smith, L Kahanpää, P Kekäläinen, W Traves, An invitation to algebraic geometry, Springer (2000) MR1788561
115 P T So’n, The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero, Proc. Amer. Math. Soc. 139 (2011) 1 MR2729065
116 P T So’n, An explicit bound for the Łojasiewicz exponent of real polynomials, Kodai Math. J. 35 (2012) 311 MR2951259
117 S Spodzieja, The Łojasiewicz exponent of subanalytic sets, Ann. Polon. Math. 87 (2005) 247 MR2208551
118 S Tan, S S T Yau, H Zuo, Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity, Proc. Amer. Math. Soc. 138 (2010) 3975 MR2679619
119 B Teissier, Variétés polaires, I : Invariants polaires des singularités d’hypersurfaces, Invent. Math. 40 (1977) 267 MR470246
120 O Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. 22 (1989) 1 MR985852
121 O E Villamayor U., Patching local uniformizations, Ann. Sci. École Norm. Sup. 25 (1992) 629 MR1198092
122 J Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779 MR2163383
123 J Włodarczyk, Resolution of singularities of analytic spaces, from: "Proceedings of Gökova Geometry–Topology Conference 2008" (editors S Akbulut, T Önder, R J Stern), GGT (2009) 31 MR2500573