International Mathematics Research Papers

Lin, Huaxin

doi: 10.1093/imrp/rpn007pmid: N/A

Abstract Let X be a compact metric space and let Λ be a ℤk (k ≥ 1) action on X. We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product C(X) ⋊Λ ℤk can be embedded into a unital simple AF-algebra if and only if X admits a strictly positive Λ-invariant Borel probability measure. Let C be a unital AH-algebra, let G be a finitely generated abelian group and let Λ : G → Aut(C) be a monomorphism. We show that C ⋊Λ G can be embedded into a unital simple AF-algebra if and only if C admits a faithful Λ-invariant tracial state. References 1 Blackadar B.,  Handelman D..  Dimension functions and traces on C*-algebras,  Journal of Functional Analysis ,  1982, vol.  45 (pg.  297- 340) Google Scholar CrossRef Search ADS   2 Bratteli O.,  Elliott G. A.,  Evans D.,  Kishimoto A..  Homotopy of a pair of approximately commuting unitaries in a simple C*-algebra,  Journal of Functional Analysis ,  1998, vol.  160 (pg.  466- 523) Google Scholar CrossRef Search ADS   3 Brown L. G.,  Douglas R. G.,  Fillmore P. A..  Unitary Equivalence Modulo the Compact Operators and Extensions of C*-Algebras,  Proceedings of a Conference on Operator Theory ,  1973 Berlin Springer(pg.  58- 128) Lecture Notes in Mathematics 345 4 Brown L. G.,  Douglas R. G.,  Fillmore P. A..  Extensions of C*-algebras and K-homology,  Annals of Mathematics ,  1977, vol.  105 (pg.  265- 324) Google Scholar CrossRef Search ADS   5 Blackadar B.,  Kirchberg E..  Generalized inductive limits of finite-dimensional C*-algebras,  Mathematische Annalen ,  1997, vol.  307 (pg.  343- 80) Google Scholar CrossRef Search ADS   6 Blackadar B.,  Kirchberg E..  Inner quasidiagonality and strong NF algebras,  Pacific Journal of Mathematics ,  2001, vol.  198 (pg.  307- 29) Google Scholar CrossRef Search ADS   7 Brown L. G.,  Dădărlat M..  Extensions of C*-algebras and quasidiagonality,  Journal of the London Mathematical Society ,  1996, vol.  53 (pg.  582- 600) Google Scholar CrossRef Search ADS   8 Brown N..  AF embeddability of crossed products of AF algebras by the integers,  Journal of Functional Analysis ,  1998, vol.  160  1(pg.  150- 75) Google Scholar CrossRef Search ADS   9 Brown N..  Crossed products of UHF algebras by some amenable groups,  Hokkaido Mathematical Journal ,  2000, vol.  29 (pg.  201- 11) Google Scholar CrossRef Search ADS   10 Brown N..  On Quasidiagonal C*-Algebras,  Operator Algebras and Applications ,  2004 Tokyo Mathematical Society of Japan(pg.  19- 64) Advanced Studies in Pure Mathematics 38 11 Brown N.,  Dadarlat M..  Extensions of Quasidiagonal C*-Algebras, K-Theory,  Operator Algebras and Applications ,  2004 Tokyo Mathematical Society of Japan(pg.  65- 84) Advanced Studies in Pure Mathematics 38 12 Choi M. D.,  Effros E..  The completely positive lifting problem for C*-Algebras,  Annals of Mathematics ,  1976, vol.  104 (pg.  309- 22) Google Scholar CrossRef Search ADS   13 Dădărlat M..  Quasidiagonal morphisms and homotopy,  Journal of Functional Analysis ,  1997, vol.  151 (pg.  213- 33) Google Scholar CrossRef Search ADS   14 Dădărlat M..  On the approximation of quasidiagonal C*-algebras,  Journal of Functional Analysis ,  1999, vol.  167 (pg.  69- 78) Google Scholar CrossRef Search ADS   15 Dădărlat M..  Residually finite dimensional C*-algebras and subquotients of the CAR algebra,  Mathematical Research Letters ,  2001, vol.  8 (pg.  545- 55) Google Scholar CrossRef Search ADS   16 Dadarlat M.,  Loring T..  A universal multicoefficient theorem for the Kasparov groups,  Duke Mathematical Journal ,  1996, vol.  84 (pg.  355- 77) Google Scholar CrossRef Search ADS   17 Davidson K.,  Herrero D.,  Salinas N..  Quasidiagonal operators, approximation, and C*-algebras,  Indiana University Mathematics Journal ,  1989, vol.  38 (pg.  973- 98) Google Scholar CrossRef Search ADS   18 Eilers S.,  Loring T.,  Pedersen G. K..  Quasidiagonal extensions and AF algebras,  Mathematische Annalen ,  1998, vol.  311 (pg.  233- 49) Google Scholar CrossRef Search ADS   19 Elliott G. A..  On the classification of C*-algebras of real rank zero,  Journal fÅr die reine und angewandte Mathematik ,  1993, vol.  443 (pg.  179- 219) 20 Elliott G. A.,  Gong G..  On the classification of C*-algebras of real rank zero. 2,  Annals of Mathematics ,  1996, vol.  144 (pg.  497- 610) Google Scholar CrossRef Search ADS   21 Elliott G. A.,  Gong G.,  Li L..  Injectivity of the connecting maps in AH inductive limit systems,  Canadian Mathematical Bulletin ,  2005, vol.  48 (pg.  50- 68) Google Scholar CrossRef Search ADS   22 Elliott G. A.,  Rørdam M..  Classification of certain infinite simple C*-Algebras, 2,  Commentarii Mathematici Helvetici ,  1995, vol.  70 (pg.  615- 38) Google Scholar CrossRef Search ADS   23 Exel R..  The soft torus and applications to almost commuting matrices,  Pacific Journal of Mathematics ,  1993, vol.  160 (pg.  207- 17) Google Scholar CrossRef Search ADS   24 Gong G.,  Lin H..  Almost multiplicative morphisms and K-theory,  International Journal of Mathematics ,  2000, vol.  11 (pg.  983- 1000) Google Scholar CrossRef Search ADS   25 Green P..  The structure of imprimitivity algebras,  Journal of Functional Analysis ,  1980, vol.  36 (pg.  88- 104) Google Scholar CrossRef Search ADS   26 Hadwin D..  Strongly quasidiagonal C*-algebras,  Journal of Operator Theory ,  1987, vol.  18 (pg.  3- 18) 27 Halmos P. R..  Quasitriangular operators,  Acta Scientiarum Mathematicarum (Szeged) ,  1968, vol.  29 (pg.  283- 93) 28 Hu S.,  Lin H.,  Xue Y..  Limits of homomorphisms with finite-dimensional range,  International Journal of Mathematics ,  2005, vol.  16 (pg.  807- 21) Google Scholar CrossRef Search ADS   29 Kishimoto A.,  Kumjian A..  The Ext class of an approximately inner automorphism,  Transactions of the American Mathematical Society ,  1998, vol.  350 (pg.  4127- 48) Google Scholar CrossRef Search ADS   30 Kishimoto A.,  Kumjian A..  The Ext class of an approximately inner automorphism. 2,  Journal of Operator Theory ,  2001, vol.  46 (pg.  99- 122) 31 Li L..  Classification of simple C*-algebras: inductive limits of matrix algebras over one-dimensional spaces,  Journal of Functional Analysis ,  2002, vol.  192 (pg.  1- 51) Google Scholar CrossRef Search ADS   32 Lin H..  Exponential rank of C*-algebras with real rank zero and the Brown–Pedersen conjectures,  Journal of Functional Analysis ,  1993, vol.  114 (pg.  1- 11) Google Scholar CrossRef Search ADS   33 Lin H..  Tracially AF C*-algebras,  Transactions of the American Mathematical Society ,  2001, vol.  353 (pg.  693- 722) Google Scholar CrossRef Search ADS   34 Lin H..  Tracial topological ranks of C*-Algebras,  Proceedings of the London Mathematical Society ,  2001, vol.  83 (pg.  199- 234) Google Scholar CrossRef Search ADS   35 Lin H.. ,  An Introduction to the Classification of Amenable C*-Algebras ,  2001 River Edge, NJ World Scientific 36 Lin H..  Embedding an AH-algebra into a simple C*-algebra with prescribed KK-data,  K-Theory ,  2001, vol.  24 (pg.  135- 56) Google Scholar CrossRef Search ADS   37 Lin H..  Residually finite dimensional and AF-embeddable C*-algebras,  Proceedings of the American Mathematical Society ,  2001, vol.  129 (pg.  1689- 96) Google Scholar CrossRef Search ADS   38 Lin H..  Classification of simple C*-algebras and higher dimensional noncommutative tori,  Annals of Mathematics 2 ,  2003, vol.  157  2(pg.  521- 44) Google Scholar CrossRef Search ADS   39 Lin H..  Simple AH-algebras of real rank zero,  Proceedings of the American Mathematical Society ,  2003, vol.  131 (pg.  3813- 19) Google Scholar CrossRef Search ADS   40 Lin H..  Classification of simple C*-algebras with tracial topological rank zero,  Duke Mathematical Journal ,  2004, vol.  125 (pg.  91- 119) Google Scholar CrossRef Search ADS   41 Lin H..  Traces and simple C*-algebras with tracial topological rank zero,  Journal für die reine und angewandte Mathematik ,  2004, vol.  568 (pg.  99- 137) 42 Lin H..  Minimal homeomorphisms and approximate conjugacy in measure,  2005  preprint arXiv/math.OA/0501262 43 Lin H..  An approximate universal coefficient theorem,  Transactions of the American Mathematical Society ,  2005, vol.  357 (pg.  3375- 405) Google Scholar CrossRef Search ADS   44 Lin H..  Classification of homomorphisms and dynamical systems,  Transactions of the American Mathematical Society ,  2007, vol.  359 (pg.  859- 95) Google Scholar CrossRef Search ADS   45 Lin H..  Unitary equivalences for essential extensions of C*-algebras,  2004  preprint arxiv.org/math.OA/0403236 46 Lin H..  Embedding crossed products into a unital simple AF-algebra,  2006  preprint arxiv.org/OA/0604047 47 Lin H..  Approximate homotopy of homomorphisms from C(X) into a simple C*-algebra,  2006  preprint arxiv.org/OA/0612125 48 Lin H..  AF-embedding of crossed products of AH-algebras by ℤ and asymptotic AF-embedding,  Indiana University Mathematics Journal ,  2008, vol.  57 (pg.  891- 944) Google Scholar CrossRef Search ADS   49 Lin H..  Asymptotic unitary equivalence and asymptotic inner automorphism,  2007  preprint arxiv.org/OA/0703610 50 Loring T..  K-theory and asymptotically commuting matrices,  Canadian Journal of Mathematics ,  1988, vol.  40 (pg.  197- 216) Google Scholar CrossRef Search ADS   51 Matui H..  AF embeddability of crossed products of AT algebras by the integers and its application,  Journal of Functional Analysis ,  2002, vol.  192 (pg.  562- 80) Google Scholar CrossRef Search ADS   52 Pasnicu C.,  Phillips N. C..  Crossed products by Z with Rokhlin property 53 Pimsner M..  Embedding some transformation group C*-algebras into AF-algebras,  Ergodic Theory and Dynamical Systems ,  1983, vol.  3 (pg.  613- 26) Google Scholar CrossRef Search ADS   54 Popa S..  On local finite-dimensional approximation of C*-algebras,  Pacific Journal of Mathematics ,  1997, vol.  181 (pg.  141- 58) Google Scholar CrossRef Search ADS   55 Rørdam M..  On the structure of simple C*-algebras tensored with a UHF-algebra. 2,  Journal of Functional Analysis ,  1992, vol.  107 (pg.  255- 69) Google Scholar CrossRef Search ADS   56 Rosenberg J..  Appendix to strongly quasidiagonal C*-Algebras,  Journal of Operator Theory ,  1987, vol.  18 (pg.  3- 18) 57 Salinas N..  Relative quasidiagonality and KK-theory,  Houston Journal of Mathematics ,  1992, vol.  18 (pg.  97- 116) 58 Thayer F. J..  Quasidiagonal C*-Algebras,  Journal of Functional Analysis ,  1977, vol.  25 (pg.  50- 7) Google Scholar CrossRef Search ADS   59 Rosenberg J.,  Schochet C..  The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor,  Duke Mathematical Journal ,  1987, vol.  55 (pg.  431- 74) Google Scholar CrossRef Search ADS   60 Toms A..  On the classification problem for nuclear C*-algebras,  2005  preprint math.OA/0509103 61 Villadsen J..  On the stable rank of simple C*-algebras,  Journal of the American Mathematical Society ,  1999, vol.  12 (pg.  1091- 102) Google Scholar CrossRef Search ADS   62 Voiculescu D..  A non-commutative Weyl- von Nuemann theorem,  Revue Roumaine de MathÅÈmatiques Pures et AppliquÅÈes ,  1976, vol.  21 (pg.  97- 113) 63 Voiculescu D..  Almost inductive limit automorphisms and embeddings into AF-algebras,  Ergodic Theory and Dynamical Systems ,  1986, vol.  6 (pg.  475- 84) Google Scholar CrossRef Search ADS   64 Voiculescu D..  A note on quasi-diagonal C*-algebras and homotopy,  Duke Mathematical Journal ,  1991, vol.  62 (pg.  267- 71) Google Scholar CrossRef Search ADS   65 Voiculescu D..  Around quasidiagonal operators,  Integral Equations Operator Theory ,  1993, vol.  17 (pg.  137- 49) Google Scholar CrossRef Search ADS   66 Zhang S..  K1-groups, quasidiagonality, and interpolation by multiplier projections,  Transactions of the American Mathematical Society ,  1991, vol.  325 (pg.  793- 818) © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press

Cumino, C.;Esteves, E.;Gatto, L.

doi: 10.1093/imrp/rpn001pmid: N/A

Abstract Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result describes all limits on C of special Weierstrass points along smooth curves degenerating to C. As an application, we recover in a unified and conceptually simpler way the computations made by Diaz and Cukierman of divisor classes of curves with special Weierstrass points in the moduli space of stable curves. In our approach there are no multiplicity issues, an usual nuisance of the method of test curves. References 1 Arbarello E.,  Cornalba M..  The Picard groups of the moduli spaces of curves.,  Topology ,  1987, vol.  26  2(pg.  153- 71) Google Scholar CrossRef Search ADS   2 CoCoATeam.  CoCoA: A system for doing Computations in Commutative Algebra  http://cocoa.dima.unige.it 3 Cukierman F..  Families of Weierstrass points.,  Duke Mathematical Journal ,  1989, vol.  58  2(pg.  317- 46) Google Scholar CrossRef Search ADS   4 Cumino C.,  Esteves E.,  Gatto L..  Special ramification loci on the double product of a general curve,  Quarterly Journal of Mathematics ,  2007  preprint arXiv:math.AG/0701662. 5 Diaz S..  Tangent spaces in moduli via deformation with applications to Weierstrass points.,  Duke Mathematical Journal ,  1984, vol.  51  4(pg.  905- 22) Google Scholar CrossRef Search ADS   6 Diaz S.. ,  Exceptional Weierstrass Points and the Divisor on Moduli Space that they Define ,  1985 Memoires of the American Mathematical Society 56, no. 327. Providence, RI American Mathematical Society 7 Eisenbud D.,  Harris J..  Limit linear series: Basic theory.,  Inventiones Mathematicae ,  1986, vol.  85  2(pg.  337- 71) Google Scholar CrossRef Search ADS   8 Eisenbud D.,  Harris J..  Existence, decomposition, and limits of certain Weierstrass points.,  Inventiones Mathematicae ,  1987, vol.  87  3(pg.  495- 515) Google Scholar CrossRef Search ADS   9 Esteves E..  Wronski algebra systems on families of singular curves.,  Annales Scientifiques de l'École Normale Supérieure (4) ,  1996, vol.  29  1(pg.  107- 34) 10 Esteves E..  “Linear systems and ramification points on reducible nodal curves.” Special issue,  Matemática Contemporanea ,  1998, vol.  14 (pg.  21- 35) 11 Esteves E.,  Gatto L..  A geometric interpretation and a new proof of a relation by Cornalba and Harris.,  Communications in Algebra ,  2003, vol.  31  8(pg.  3753- 70) Google Scholar CrossRef Search ADS   12 Esteves E.,  Medeiros N..  Limit canonical systems on curves with two components.,  Inventiones Mathematicae ,  2002, vol.  149  2(pg.  267- 338) Google Scholar CrossRef Search ADS   13 Fulton W.. ,  Intersection Theory ,  1984 vol. 3. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin Springer 14 Gatto L..  Weight sequences versus gap sequences at singular points of Gorenstein curves.,  Geometriae Dedicata ,  1995, vol.  54  3(pg.  267- 300) Google Scholar CrossRef Search ADS   15 Gatto L..  On the closure in of smooth curves having a special Weierstrass point.,  Mathematica Scandinavica ,  2001, vol.  88  1(pg.  41- 71) Google Scholar CrossRef Search ADS   16 Gieseker D..  Lectures on Moduli of Curves,  Tata Institute of Fundamental Research, Lecture Notes. ,  1982 Heidelberg, Germany Springer 17 Harris J.,  Morrison I.. ,  Moduli of Curves. ,  1998 New York Springer Graduate Texts in Mathematics 187 18 Harris J.,  Mumford D..  On the Kodaira dimension of the moduli space of curves.,  Inventiones Mathematicae ,  1982, vol.  67 (pg.  23- 86) Google Scholar CrossRef Search ADS   19 Laksov D.,  Thorup A..  The algebra of jets.,  Michigan Mathematical Journal ,  2000, vol.  48 (pg.  393- 416) Google Scholar CrossRef Search ADS   20 Laksov D.,  Thorup A..  Wronski systems for families of local complete intersection curves.,  Communications in Algebra ,  2003, vol.  31  8(pg.  4007- 35) Google Scholar CrossRef Search ADS   21 Looijenga E..  Smooth Deligne–Mumford compactifications by means of Prym level structures.,  Journal of Algebraic Geometry ,  1994, vol.  3 (pg.  283- 93) 22 Mumford D..  Stability of projective varieties,  L'Enseignement Mathématique 2 ,  1977, vol.  23 (pg.  39- 110) 23 Mumford D..  Artin M.,  Tate J..  Towards an enumerative geometry of the moduli space of curves,  Arithmetic and Geometry ,  1983, vol.  2 (pg.  271- 328)  Progress in Mathematics 36. Boston: Birkhäuser 24 Mumford D.,  Fogarty J.. ,  Geometric Invariant Theory ,  1982 2nd ed. Berlin Springer 25 Osserman B..  A limit linear series moduli scheme.,  Annales de l'Institute Fourier (Grenoble) ,  2006, vol.  56  4(pg.  1165- 205) Google Scholar CrossRef Search ADS   26 Pikaart M.,  Jong A. J. de.  Moduli of curves with non-abelian level structure,  The Moduli Space of Curves, (pg.  483- 509)  Progress in Mathematics 129. Boston: Birkhäuser, 1995. © The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: [email protected] Oxford University Press

Yamakawa, Daisuke

doi: 10.1093/imrp/rpn008pmid: N/A

Abstract Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called a multiplicative preprojective algebra. In this paper, we study a moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there is a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parameterizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight, and associated graded local systems around each puncture. References 1 Alekseev A.,  Malkin A.,  Meinrenken E..  Lie group valued moment maps,  Journal of Differential Geometry ,  1998, vol.  48  3(pg.  445- 95) 2 Boalch P..  Stokes matrices, Poisson Lie groups and Frobenius manifolds,  Inventiones Mathematicae ,  2001, vol.  146  3(pg.  479- 506) Google Scholar CrossRef Search ADS   3 Boalch P..  Symplectic manifolds and isomonodromic deformations,  Advances in Mathematics ,  2001, vol.  163  2(pg.  137- 205) Google Scholar CrossRef Search ADS   4 Boalch P..  Quasi-Hamiltonian geometry of meromorphic connections,  Duke Mathematical Journal ,  2007, vol.  139  2(pg.  369- 405) Google Scholar CrossRef Search ADS   5 Borel A.. ,  Linear Algebraic Groups ,  1991 2nd ed New York Springer Graduate Texts in Mathematics 126 6 Bott R.,  Tu L. W.. ,  Differential Forms in Algebraic Topology ,  1982 New York Springer Graduate Texts in Mathematics 82 7 Cassens H.,  Slodowy P..  On Kleinian Singularities and Quivers,  Singularities ,  1998 Basel, Switzerland Birkhauser(pg.  263- 88) Progress in Mathematics 162 8 Crawley-Boevey W..  Geometry of the moment map for representations of quivers,  Compositio Mathematicae ,  2001, vol.  126  3(pg.  257- 93) Google Scholar CrossRef Search ADS   9 Crawley-Boevey W..  Normality of Marsden Weinstein reductions for representations of quivers,  Mathematische Annalen ,  2003, vol.  325  1(pg.  55- 79) Google Scholar CrossRef Search ADS   10 Crawley-Boevey W..  On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero,  Duke Mathematical Journal ,  2003, vol.  118  2(pg.  339- 52) Google Scholar CrossRef Search ADS   11 Crawley-Boevey W..  Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity,  Publications Mathématiques. Institut de Hautes Études Scientifiques ,  2004, vol.  100 (pg.  171- 207) Google Scholar CrossRef Search ADS   12 Crawley-Boevey W.,  Shaw P..  Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem,  Advances in Mathematics ,  2006, vol.  201  1(pg.  180- 208) Google Scholar CrossRef Search ADS   13 Crawley-Boevey W.,  Van den Bergh M..  “Absolutely indecomposable representations and Kac-Moody Lie algebras.” With an appendix by Hiraku Nakajima,  Inventiones Mathematicae ,  2004, vol.  155  3(pg.  537- 59) Google Scholar CrossRef Search ADS   14 Deligne P.. ,  Équations différentielles à points singuliers réguliers ,  1970 Berlin Springer Lecture Notes in Mathematics 163 15 Dettweiler M.,  Reiter S..  An algorithm of Katz and its application to the inverse Galois problem,  Journal of Symbolic Computation ,  2000, vol.  30  6(pg.  761- 98) Google Scholar CrossRef Search ADS   16 Grauert H.,  Remmert R.. ,  Coherent Analytic Sheaves ,  1984 Berlin Springer Grundlehren der Mathematischen Wissenschaften 265 [Fundamental Principles of Mathematical Sciences] 17 Hausel T..  Cohomology of hyperkähler manifolds via arithmetic harmonic analysis  A talk at Kyoto University, 2005. http://www2.maths.ox.ac.uk/~hausel/talks.html 18 Hausel T.,  Rodriguez-Villegas F..  Mixed Hodge polynomials of character varieties,  Inventiones Mathematicae ,  2006 19 Hitchin N..  “Frobenius Manifolds.” With notes by David Calderbank,  Gauge Theory and Symplectic Geometry ,  1997 Dorderecht, the Netherlands Kluwer(pg.  69- 112) NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 488 20 Inaba M..  Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence,  2006  preprint arXiv:math.AG/0602004 21 Inaba M.,  Iwasaki K.,  Saito M..  Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equation of type 6: 1,  Kyoto University, Research Institute for Mathematical Sciences Publications ,  2006, vol.  42  4(pg.  987- 1089) Google Scholar CrossRef Search ADS   22 Kac V. G.. ,  Infinite-dimensional Lie algebras ,  1990 3rd ed Cambridge Cambridge University Press 23 King A. D..  Moduli of representations of finite-dimensional algebras,  Quarterly Journal of Mathematics ,  1994, vol.  45  180(pg.  515- 30) Google Scholar CrossRef Search ADS   24 Kodaira K.. ,  Complex Manifolds and Deformation of Complex Structures ,  2005 Berlin Springer Classics in Mathematics 25 Kraft H.,  Procesi C..  Closures of conjugacy classes of matrices are normal,  Inventiones Mathematicae ,  1979, vol.  53  3(pg.  227- 47) Google Scholar CrossRef Search ADS   26 Kronheimer P. B..  The construction of ALE spaces as hyper-Kahler quotients,  Journal of Differential Geometry ,  1989, vol.  29  3(pg.  665- 83) 27 Kronheimer P. B.,  Nakajima H..  Yang-Mills instantons on ALE gravitational instantons,  Mathematische Annalen ,  1990, vol.  288  2(pg.  263- 307) Google Scholar CrossRef Search ADS   28 Lusztig G..  On quiver varieties,  Advances in Mathematics ,  1998, vol.  136  1(pg.  141- 82) Google Scholar CrossRef Search ADS   29 Lusztig G..  Quiver varieties and Weyl group actions,  Annales de l'Institut Fourier ,  2000, vol.  50  2(pg.  461- 89) Google Scholar CrossRef Search ADS   30 Maffei A..  A remark on quiver varieties and Weyl groups,  Annali della Scuola Normale Superiore di Pisa: Classe di Scienze ,  2002, vol.  51  3(pg.  649- 86) 31 Marsden J.,  Weinstein A..  Reduction of symplectic manifolds with symmetry,  Reports on Mathematical Physics ,  1974, vol.  5  1(pg.  121- 30) Google Scholar CrossRef Search ADS   32 Nakajima H..  Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras,  Duke Mathematical Journal ,  1994, vol.  76  2(pg.  365- 416) Google Scholar CrossRef Search ADS   33 Nakajima H..  Quiver varieties and Kac-Moody algebras,  Duke Mathematical Journal ,  1998, vol.  91  3(pg.  515- 60) Google Scholar CrossRef Search ADS   34 Nakajima H..  Quiver varieties and finite-dimensional representations of quantum affine algebras,  Journal of the American Mathematical Society ,  2001, vol.  14  1(pg.  145- 238) Google Scholar CrossRef Search ADS   35 Nakajima H..  Reflection functors for quiver varieties and Weyl group actions,  Mathematische Annalen ,  2003, vol.  327  4(pg.  671- 721) Google Scholar CrossRef Search ADS   36 Newstead P. E.. ,  Introduction to Moduli Problems and Orbit Spaces ,  1978 Bombay Tata Institute of Fundamental Research Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51 37 Simpson C. T..  Harmonic bundles on noncompact curves,  Journal of the American Mathematical Society ,  1990, vol.  3  3(pg.  713- 70) Google Scholar CrossRef Search ADS   38 Sjamaar R..  Holomorphic slices, symplectic reduction and multiplicities of representations,  Annals of Mathematics, Second Series ,  1995, vol.  141  1(pg.  87- 129) Google Scholar CrossRef Search ADS   39 Van den Bergh M..  Double Poisson algebras,  2004  preprint arXiv:math.QA/0410528 40 Bergh M. Van den.  Non-commutative quasi-Hamiltonian spaces,  2007  preprint arXiv:math.QA/0703293 © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press

Duyckaerts, Thomas;Merle, Frank

doi: 10.1093/imrp/rpn002pmid: N/A

Abstract We consider the energy-critical nonlinear focusing wave equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [8], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u0, u1) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analogous to [3], which treats the energy-critical nonlinear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions. References 1 Aubin Thierry.  Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,  Journal de Mathématiques Pures et Appliquées ,  1976, vol.  55  3(pg.  269- 96) 2 Bahouri Hajer,  Gérard Patrick.  High frequency approximation of solutions to critical nonlinear wave equations,  American Journal of Mathematics ,  1999, vol.  121  1(pg.  131- 75) Google Scholar CrossRef Search ADS   3 Duyckaerts T.,  Merle F..  Dynamic of threshold solutions for energy-critical nls,  Geometric and Functional Analysis   (forthcoming) 4 Ginibre J.,  Soffer A.,  Velo G..  The global Cauchy problem for the critical nonlinear wave equation,  Journal of Functional Analysis ,  1992, vol.  110  1(pg.  96- 130) Google Scholar CrossRef Search ADS   5 Ginibre J.,  Velo G..  Generalized Strichartz inequalities for the wave equation,  Journal of Functional Analysis ,  1995, vol.  133  1(pg.  50- 68) Google Scholar CrossRef Search ADS   6 Kapitanski Lev.  Global and unique weak solutions of nonlinear wave equations,  Mathematical Research Letters ,  1994, vol.  1  2(pg.  211- 23) Google Scholar CrossRef Search ADS   7 Kenig Carlos E.,  Merle Frank.  Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case,  Inventiones Mathematicae ,  2006, vol.  166  3(pg.  645- 75) Google Scholar CrossRef Search ADS   8 Kenig Carlos E.,  Merle Frank.  Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation,  Acta Mathematica   (forthcoming) 9 Kenig Carlos E.,  Ponce Gustavo,  Vega Luis.  Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,  Communications on Pure and Applied Mathematics ,  1993, vol.  46  4(pg.  527- 620) Google Scholar CrossRef Search ADS   10 Krieger J.,  Schlag W.,  Tataru D..  Slow blow-up solutions for the h1(R3) critical focusing semi-linear wave equation in R3,  2007  preprint arXiv.org:math/0702033 11 Lions P.-L..  The concentration-compactness principle in the calculus of variations. The limit case. II,  Revista Matematica Iberoamericana ,  1985, vol.  1  2(pg.  45- 121) Google Scholar CrossRef Search ADS   12 Lindblad Hans,  Sogge Christopher D..  On existence and scattering with minimal regularity for semilinear wave equations,  Journal of Functional Analysis ,  1995, vol.  130  2(pg.  357- 426) Google Scholar CrossRef Search ADS   13 Pecher Hartmut.  Nonlinear small data scattering for the wave and Klein-Gordon equation,  Mathematische Zeitschriftath ,  1984, vol.  185  2(pg.  261- 70) Google Scholar CrossRef Search ADS   14 Rey Olivier.  The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent,  Journal of Functional Analysis ,  1990, vol.  89  1(pg.  1- 52) Google Scholar CrossRef Search ADS   15 Reed Michael,  Simon Barry. ,  Methods of Modern Mathematical Physics. IV. Analysis of Operators ,  1978 New York Academic Press 16 Schlag W.,  Krieger J..  On the focusing critical semi-linear wave equation,  American Journal of Mathematics ,  2007, vol.  129  3(pg.  843- 913) Google Scholar CrossRef Search ADS   17 Sogge Christopher D.. ,  Lectures on Nonlinear Wave Equations ,  1995 Boston, MA International Press Monographs in Analysis 2 18 Shatah Jalal,  Struwe Michael.  Well-posedness in the energy space for semilinear wave equations with critical growth,  International Mathematics Research Notices ,  1994, vol.  7 (pg.  303- 9) Google Scholar CrossRef Search ADS   19 Shatah Jalal,  Struwe Michael. ,  Geometric Wave Equations ,  1998 New York New York University Courant Institute of Mathematical Sciences Courant Lecture Notes in Mathematics 2 20 Talenti Giorgio.  Best constant in Sobolev inequality,  Annali di Mathematica Pure ed Applicata ,  1976, vol.  4  110(pg.  353- 72) Google Scholar CrossRef Search ADS   © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press

Aptekarev, Alexander I.;Kuijlaars, Arno B. J.;Van Assche, Walter

doi: 10.1093/imrp/rpm007pmid: N/A

Abstract We investigate the asymptotic behavior for type II Hermite–Padé approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite–Padé approximants are described by an algebraic function h of order and genus 0. This situation gives rise to a vector-potential equilibrium problem for measures λ, μ1, and μ2, and the poles of the common denominator are asymptotically distributed like λ/2. We also work out the strong asymptotics for the corresponding Hermite–Padé approximants by using a 3 × 3 Riemann–Hilbert problem that characterizes this Hermite–Padé approximation problem. References 1 Abramowitz M.,  Stegun I. A.. ,  Handbook of Mathematical Functions ,  1972 10th ed New York Dover 2 Angelesco A..  Sur deux extensions des fractions continues algébriques,  Comptes Rendus de l'Académie des Sciences, Paris ,  1919, vol.  168 (pg.  262- 5) 3 Aptekarev A. I..  “Asymptotics of polynomials of simultaneous orthogonality in the Angelesco case” [in Russian],  Matematicheskii Sbornik   136 (178), no. 1 (1988): 56–84; translated in Mathematics of the USSR-Sbornik 64, no. 1 (1989): 57–84 4 Aptekarev A. I..  Multiple orthogonal polynomials,  Journal of Computational and Applied Mathematics ,  1998, vol.  99  1(pg.  423- 48) Google Scholar CrossRef Search ADS   5 Aptekarev A. I..  “Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems” [in Russian],  Matematicheskii Sbornik ,  1999, vol.  190  5(pg.  3- 44)  translated in Sbornik Mathematics 190, no. 5–6 (1999): 631–69 Google Scholar CrossRef Search ADS   6 Aptekarev A. I..  “Sharp constants for rational approximations of analytic functions” [in Russian],  Matematicheskii Sbornik ,  2002, vol.  193  1(pg.  3- 72)  translated in Sbornik Mathematics 193, no. 1–2 (2002): 1–72 Google Scholar CrossRef Search ADS   7 Aptekarev A. I.,  Bleher P. M.,  Kuijlaars A. B. J..  Large n limit of Gaussian random matrices with external source, Part 2,  Communications in Mathematical Physics ,  2005, vol.  259 (pg.  367- 89) Google Scholar CrossRef Search ADS   8 Aptekarev A. I.,  Kalyagin V. A..  “Asymptotic behavior of an nth degree root of polynomials of simultaneous orthogonality, and algebraic functions” [in Russian],  Akademiya Nauk SSSR, Institut Prikladnoi Matematiki ,  1986 60  preprint 9 Aptekarev A. I.,  Stahl H..  Gonchar A.,  Saff E.B..  Asymptotics of Hermite–Padè polynomials,  Progress in Approximation Theory ,  1992 New York Springer(pg.  127- 67) Springer Series in Computational Mathematics 19 10 Aptekarev A. I.,  Van Assche W..  Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight,  Journal of Approximation Theory ,  2004, vol.  129  2(pg.  129- 66) Google Scholar CrossRef Search ADS   11 Baik J.,  Deift P.,  McLaughlin K. T-R.,  Miller P.,  Zhou X..  Optimal tail estimates for directed last passage site percolation with geometric random variables,  Advances in Theoretical and Mathematical Physics ,  2001, vol.  5  6(pg.  1207- 50) Google Scholar CrossRef Search ADS   12 Baumel R. T.,  Gammel J. L.,  Nuttall J..  Asymptotic form of Hermite–Padé polynomials,  IMA Journal of Applied Mathematics ,  1981, vol.  27  3(pg.  335- 57) Google Scholar CrossRef Search ADS   13 Bernstein S. N..  Sur les polynomes orthogonaux relatifs a' un segment fini 1,  Journal de Mathématiques Pures et Appliquées ,  1930, vol.  9 (pg.  127- 77) 14 Bernstein S. N..  Sur les polynomes orthogonaux relatifs a' un segment fini 2,  Journal de Mathématiques Pures et Appliquées ,  1931, vol.  10 (pg.  219- 86) 15 Bleher P.,  Its A..  Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model,  Annals of Mathematics ,  1999, vol.  150 (pg.  185- 266) Google Scholar CrossRef Search ADS   16 Bleher P.,  Its A..  Double scaling limit in the random matrix model: the Riemann–Hilbert approach,  Communications in Pure and Applied Mathematics ,  2003, vol.  56 (pg.  433- 516) Google Scholar CrossRef Search ADS   17 Bleher P. M.,  Kuijlaars A. B. J..  Large n limit of Gaussian random matrices with external source, Part 1,  Communications in Mathematical Physics ,  2004, vol.  252 (pg.  43- 76) Google Scholar CrossRef Search ADS   18 Bleher P. M.,  Kuijlaars A. B. J..  Large n limit of Gaussian random matrices with external source, Part 3: Double scaling limit,  Communications in Mathematical Physics ,  2007, vol.  270 (pg.  481- 517) Google Scholar CrossRef Search ADS   19 Borwein P. B..  Quadratic Hermite–Padé approximation to the exponential function,  Constructive Approximation ,  1986, vol.  2  4(pg.  291- 302) Google Scholar CrossRef Search ADS   20 de Bruin M..  Some aspects of simultaneous rational approximation,  Numerical Analysis and Mathematical Modelling ,  1990, vol.  24  Warsaw, PL Banach Center Publications(pg.  51- 84) 21 Bustamante J.,  Lagomasino G. López.  “Hermite–Padé approximation for Nikishin systems of analytic functions” [in Russian],  Matematicheskii Sbornik ,  1992, vol.  183  11(pg.  117- 38)  translated in Russian Academy of Sciences Sbornik Mathematics 77, no. 2 (1994): 367–84 22 Claeys T.,  Kuijlaars A. B. J..  Universality of the double scaling limit in random matrix models,  Communications in Pure and Applied Mathematics ,  2006, vol.  59 (pg.  1573- 603) Google Scholar CrossRef Search ADS   23 Claeys T.,  Kuijlaars A. B. J..  Universality in unitary random matrix ensembles when the soft edge meets the hard edge,  Integrable Systems, Random Matrices, and Applications ,  2007  preprint arXiv:math-ph/0701003 24 Claeys T.,  Kuijlaars A. B. J.,  Vanlessen M..  Multi-critical unitary random matrix ensembles and the general Painlevé II equation,  Annals of Mathematics ,  2005  preprint arXiv:math-ph/0508062 25 Claeys T.,  Vanlessen M..  Universality of a double scaling limit near singular edge points in random matrix models,  Communications in Mathematical Physics ,  2007, vol.  273 (pg.  499- 532) Google Scholar CrossRef Search ADS   26 Daems E.,  Kuijlaars A. B. J.,  Veys W..  Asymptotics of non-intersecting Brownian motions and a 4 × 4 Riemann–Hilbert problem,  Journal of Approximation Theory ,  2007, vol.  146  1(pg.  91- 114) Google Scholar CrossRef Search ADS   27 Deift P.. ,  Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach ,  1999 Providence, RI American Mathematical Society Courant Lecture Notes in Mathematics 3 28 Deift P.,  Kriecherbauer T.,  McLaughlin K. T-R.,  Venakides S.,  Zhou X..  Asymptotics for polynomials orthogonal with respect to varying exponential weights,  Internatonal Mathematics Reseasch Notices ,  1997, vol.  1997  16(pg.  759- 82) Google Scholar CrossRef Search ADS   29 Deift P.,  Kriecherbauer T.,  McLaughlin K. T-R.,  Venakides S.,  Zhou X..  Uniform asymptotics for orthogonal polynomials,  Special issue, Documenta Mathematica ,  1998, vol.  3 (pg.  491- 501) 30 Deift P.,  Kriecherbauer T.,  McLaughlin K. T-R.,  Venakides S.,  Zhou X..  Strong asymptotics of orthogonal polynomials with respect to exponential weights,  Communications in Pure and Applied Mathematics ,  1999, vol.  52  12(pg.  1491- 552) Google Scholar CrossRef Search ADS   31 Deift P.,  Kriecherbauer T.,  McLaughlin K. T-R.,  Venakides S.,  Zhou X..  Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory,  Communications in Pure and Applied Mathematics ,  1999, vol.  52  11(pg.  1335- 425) Google Scholar CrossRef Search ADS   32 Deift P.,  Zhou X..  A steepest descent method for oscillatory Riemann–Hilbert problems: Asymptotics for the MKdV equation,  Annals of Mathematics ,  1993, vol.  137  2(pg.  295- 368) Google Scholar CrossRef Search ADS   33 Duits M.,  Kuijlaars A. B. J..  Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight,  Nonlinearity ,  2006, vol.  19 (pg.  2211- 45) Google Scholar CrossRef Search ADS   34 Driver K. A..  Non-diagonal quadratic Hermite–Padé approximation to the exponential function,  Journal of Computational and Applied Mathematics ,  1995, vol.  65 (pg.  125- 34) Google Scholar CrossRef Search ADS   35 Driver K.,  Stahl H..  Simultaneous rational approximants to Nikishin systems 1,  Acta Scientiarum Mathematicarum ,  1995, vol.  60  1–2(pg.  245- 63) 36 Driver K.,  Stahl H..  Simultaneous rational approximants to Nikishin systems 2,  Acta Scientiarum Mathematicarum ,  1995, vol.  61  1–4(pg.  261- 84) 37 Driver K. A.,  Temme N. M..  On polynomials related with Hermite–Padé approximations to the exponential function,  Journal of Approximation Theory ,  1998, vol.  95 (pg.  101- 22) Google Scholar CrossRef Search ADS   38 Fokas A. S.,  Its A. R.,  Kitaev A. V..  An isomonodromy approach to the theory of two-dimensional quantum gravity,  Uspekhi Matematicheskikh Nauk ,  1990, vol.  45  6(pg.  135- 6)  translated in Russian Mathematical Surveys 45, no. 6 (1990): 155–7 39 Fokas A. S.,  Its A. R.,  Kitaev A. V..  The isomonodromy approach to matrix models in 2D quantum gravity,  Communications in Mathematical Physics ,  1992, vol.  147  2(pg.  395- 430) Google Scholar CrossRef Search ADS   40 Gammel J. L.,  Nuttall J..  Note on generalized Jacobi polynomials,  The Riemann Problem: Complete Integrability and Arithmetic Applications ,  1982 Berlin Springer(pg.  258- 70) Lecture Notes in Mathematics 925 41 Gonchar A. A.,  Rakhmanov E. A..  On the convergence of simultaneous Padé approximants for systems of functions of Markov type,  Trudy Matematicheskogo Instituta imeni V.A. Steklov ,  1981, vol.  157 (pg.  31- 48) 42 Gonchar A. A.,  Rakhmanov E. A..  On the equilibrium problem for vector potentials,  Uspekhi Matematicheskikh Nauk ,  1985, vol.  40  4(pg.  155- 6)  translated in Russian Mathematical Surveys 40, no. 4 (1985): 183–4 43 Gonchar A. A.,  Rakhmanov E. A.,  Sorokin V. N..  On Hermite–Padé approximants for systems of functions of Markov type,  Matematicheskii Sbornik ,  1997, vol.  188  5(pg.  33- 58)  translated in Sbornik Mathematics 188, no. 5 (1997): 671–96 Google Scholar CrossRef Search ADS   44 Hermite C..  Sur la fonction exponentielle 1  Comptes Rendus de l'Académie des Sciences, Paris 77 (1873): 18–24 45 Hermite C..  Sur la fonction exponentielle 2  Comptes Rendus de l'Académie des Sciences, Paris 77 (1873): 74–9 46 Hermite C..  Sur la fonction exponentielle 3  Comptes Rendus de l'Académie des Sciences, Paris 77 (1873): 226–33 47 Its A. R.,  Kuijlaars A. B. J.,  Ostensson J..  Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent,  2007  preprint arXiv:0704.1972 48 Kalyagin V. A..  On a class of polynomials defined by two orthogonality relations,  Matematicheskii Sbornik   110 (152), no. 4 (1979): 609–27 49 Kamvissis S.,  McLaughlin K. T-R.,  Miller P. D.. ,  Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation ,  2003 Princeton, NJ Princeton University Press Annals of Mathematics Studies 154 50 Kriecherbauer T.,  McLaughlin K. T-R..  Strong asymptotics of polynomials orthogonal with respect to Freud weights,  International Mathematical Research Notices ,  1999, vol.  1999  6(pg.  299- 333) Google Scholar CrossRef Search ADS   51 Kuijlaars A. B. J.,  McLaughlin K. T-R.,  Van Assche W.,  Vanlessen M..  The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [− 1, 1],  Advances in Mathematics ,  2004, vol.  188  2(pg.  337- 98) Google Scholar CrossRef Search ADS   52 Kuijlaars A. B. J.,  Stahl H.,  Van Assche W.,  Wielonsky F..  Asymptotique des approximants de Hermite–Padé quadratiques de la fonction exponentielle et problèmes de Riemann–Hilbert,  Comptes Rendus de l'Academie des Sciences ,  2003, vol.  336  1(pg.  893- 6) 53 Kuijlaars A. B. J.,  Stahl H.,  Van Assche W.,  Wielonsky F..  Type II Hermite–Padé approximation to the exponential function,  Journal of Computational and Applied Mathematics ,  2007, vol.  207  2(pg.  227- 44) Google Scholar CrossRef Search ADS   54 Kuijlaars A. B. J.,  Van Assche W.,  Wielonsky F..  Quadratic Hermite–Padé approximation to the exponential function: a Riemann–Hilbert approach,  Constructive Approximation ,  2005, vol.  21  3(pg.  351- 412) Google Scholar CrossRef Search ADS   55 Lysov V. G..  Strong asymptotics for the Hermite–Padé approximants for a system of Stieltjes functions with Laguerre weights,  Matematicheskii Sbornik ,  2005, vol.  196  12(pg.  99- 122) Google Scholar CrossRef Search ADS   56 Lysov V. G.,  Wielonsky F..  Strong asymptotics for multiple Laguerre polynomials,  Constructive Approximation ,  2008, vol.  28  1(pg.  61- 111) Google Scholar CrossRef Search ADS   57 Mahler K..  Perfect systems,  Compositio Mathematica ,  1968, vol.  19 (pg.  95- 166) 58 Markov A. A..  Deux demonstrations de la convergence de certaines fractions continues,  Acta Mathematica ,  1895, vol.  19 (pg.  93- 104) Google Scholar CrossRef Search ADS   59 Nikishin E. M..  A system of Markov functions,  Vestnik Moskovskogo Universiteta Seriya 1, Matematika Mekhanika ,  1979, vol.  34  4(pg.  60- 3)  translated in Moscow University Mathematics Bulletin 34, no. 4 (1979): 63–6 60 Nikishin E. M..  Simultaneous Padé approximants,  Matematicheskii Sbornik   113 (155), no. 4 (1980): 499–519 61 Nikishin E. M..  Asymptotic behavior of linear forms for simultaneous Padé approximants,  Izvestiya Vysshikh Uchebnykh Zavedenii Matematika ,  1986 2(pg.  33- 41)  translated in Soviet Mathematics 30, no. 2 (1986): 43–52 62 Nikishin E. M.,  Sorokin V. N..  Rational Approximations and Orthogonality,  Translations of Mathematical Monographs ,  1991, vol.  92  Providence, RI American Mathematical Society 63 Nuttall J..  Saff E. B.,  Varga R. S..  The convergence of Padé approximants to functions with branch points,  Padé and Rational Approximation ,  1977 New York Academic Press(pg.  101- 9) 64 Nuttall J..  Bardos C.,  Bessis D..  Sets of minimum capacity: Padé approximants and the bubble problem,  Bifurcation Phenomena in Mathematical Physics and Related Topics ,  1980 Dordrecht, DE Reidel(pg.  185- 201) 65 Nuttall J..  Hermite–Padé approximants to functions meromorphic on a Riemann surface,  Journal of Approximation Theory ,  1981, vol.  32  3(pg.  233- 40) Google Scholar CrossRef Search ADS   66 Nuttall J..  Asymptotics of diagonal Hermite–Padé polynomials,  Journal of Approximation Theory ,  1984, vol.  42  4(pg.  299- 386) Google Scholar CrossRef Search ADS   67 Nuttall J.,  Singh S. R..  Orthogonal polynomials and Padé approximants associated with a system of arcs,  Journal of Approximation Theory ,  1977, vol.  21  1(pg.  1- 42) Google Scholar CrossRef Search ADS   68 Saff E. B.,  Totik V.. ,  Logarithmic Potentials with External Fields ,  1997 New York Springer 69 Siegel C. L.. ,  Topics in Complex Function Theory ,  1969, vol.  1  New York Interscience 70 Siegel C. L.. ,  Topics in Complex Function Theory ,  1971, vol.  2  New York Interscience 71 Stahl H..  The structure of extremal domains associated with an analytic function,  Complex Variables Theory and Applications ,  1985, vol.  4  4(pg.  339- 54) Google Scholar CrossRef Search ADS   72 Stahl H..  Orthogonal polynomials with complex-valued weight function 1,  Constructive Approximation ,  1986, vol.  2  3(pg.  225- 40) Google Scholar CrossRef Search ADS   73 Stahl H..  Orthogonal polynomials with complex-valued weight function 2,  Constructive Approximation ,  1986, vol.  2  3(pg.  241- 51) Google Scholar CrossRef Search ADS   74 Stahl H..  Simultaneous rational approximants,  Computational Methods and Function Theory ,  1995 Singapore World Scientific(pg.  325- 49) 75 Stahl H..  Asymptotics for quadratic Hermite–Padé polynomials associated with the exponential function,  Electronic Transactions on Numerical Analysis ,  2002, vol.  14 (pg.  195- 222) 76 Stahl H..  Quadratic Hermite–Padé polynomials associated with the exponential function,  Journal of Approximation Theory ,  2003, vol.  125 (pg.  238- 94) Google Scholar CrossRef Search ADS   77 Stahl H..  Asymptotic distributions of zeros of quadratic Hermite–Padé polynomials associated with the exponential function,  Constructive Approximation ,  2006, vol.  23  2(pg.  121- 64) Google Scholar CrossRef Search ADS   78 Szegő G..  Orthogonal Polynomials,  American Mathematical Society ,  1975 3rd ed Providence, RI Colloquium Publications 79 Van Assche W..  Padé and Hermite–Padé approximation and orthogonality,  Surveys in Approximation Theory ,  2006, vol.  2 (pg.  61- 91) 80 Van Assche W.,  Geronimo J. S.,  Kuijlaars A. B. J..  Bustoz J., et al.  Riemann–Hilbert problems for multiple orthogonal polynomials,  Special Functions 2000: Current Perspective and Future Directions ,  2001 Dordrecht Kluwer Academic(pg.  23- 59) NATO Science Series 2: Mathematics, Physics and Chemistry 30 81 Wielonsky F..  Asymptotics of diagonal Hermite–Padé approximants to ez,  Journal of Approximation Theory ,  1997, vol.  90 (pg.  283- 98) Google Scholar CrossRef Search ADS   82 Wielonsky F..  Berndt B. C.,  Gesztesy F..  Some properties of Hermite–Padé approximants to ez,  Continued Fractions: From Analytic Number Theory to Constructive Approximation ,  1999 Providence, RI American Mathematical Society(pg.  369- 79) Contemporary Mathematics 236 © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press

Adler, Jeffrey D.;Spice, Loren

doi: 10.1093/imrp/rpn003pmid: N/A

Abstract We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations. References 1 Adler J. D..  Refined anisotropic K-types and supercuspidal representations,  Pacific Journal of Mathematics ,  1998, vol.  185 (pg.  1- 32) Google Scholar CrossRef Search ADS   2 Adler J. D.,  Corwin L.,  Sally P. J.Jr.  Discrete series characters of division algebras and GLn over a p-adic field,  Contributions to Automorphic Forms, Geometry, and Number Theory ,  2004 Baltimore, MD Johns Hopkins University Press(pg.  57- 64) 3 Adler J. D.,  DeBacker S..  Some applications of Bruhat–Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group,  Michigan Mathematical Journal ,  2002, vol.  50 (pg.  263- 86) Google Scholar CrossRef Search ADS   4 Adler J. D.,  DeBacker S..  Murnaghan–Kirillov theory for supercuspidal representations of tame general linear groups,  Journal für die Reine und Angewandte Mathematik ,  2004, vol.  575 (pg.  1- 35) Google Scholar CrossRef Search ADS   5 Adler J. D.,  Spice L..  Supercuspidal characters of reductive p-adic groups,  2007  preprint arXiv:0707.3313 6 Boller J..  Characters of some supercuspidal representations of p-adic Sp4(F),  1999  PhD Thesis, The University of Chicago 7 Borel A.. ,  Linear Algebraic Groups ,  1991 New York Springer Graduate Texts in Mathematics 126 8 Borel A.,  Springer T. A..  Rationality properties of linear algebraic groups 2,  Tôhoku Mathematical Journal ,  1968, vol.  20  no. 2(pg.  443- 97) Google Scholar CrossRef Search ADS   9 Borel A.,  Tits J..  Groupes réductifs,  Publications Mathématiques de l'Institut des Hautes Études Scientifiques ,  1965, vol.  27 (pg.  55- 150) Google Scholar CrossRef Search ADS   10 Borel A.,  Tits J..  Homomorphismes ‘abstraits’ de groupes algébriques simples,  Annals of Mathematics ,  1973, vol.  97  no. 2(pg.  499- 571) Google Scholar CrossRef Search ADS   11 Bourbaki N..  Lie Groups and Lie Algebras,  2002  Chap. 4–6, Elements of Mathematics. Berlin: Springer 12 Bruhat F.,  Tits J..  Groupes réductifs sur un corps local,  Publications Mathématiques de l'Institut des Hautes Études Scientifiques ,  1972, vol.  41 (pg.  5- 251) Google Scholar CrossRef Search ADS   13 Bruhat F.,  Tits J..  Groupes réductifs sur un corps local 2: Schémas en groupes. Existence d'une donnée radicielle valuée,  Publications Mathématiques de l'Institut des Hautes Études Scientifiques ,  1984, vol.  60 (pg.  197- 376) 14 Corwin L.,  Moy A.,  Sally P. J.Jr.  Supercuspidal character formulas for ,  Representation theory and harmonic analysis ,  1995 Providence, RI American Mathematical Society(pg.  1- 11) Contemporary Mathematics 191 15 DeBacker S..  On supercuspidal characters of GLℓ, ℓ a prime,  1997  PhD Thesis, The University of Chicago 16 DeBacker S..  Some applications of Bruhat–Tits theory to harmonic analysis on a reductive p-adic group,  Michigan Mathematical Journal ,  2002, vol.  50 (pg.  241- 61) Google Scholar CrossRef Search ADS   17 DeBacker S..  Parametrizing nilpotent orbits via Bruhat–Tits theory,  Annals of Mathematics ,  2002, vol.  156  no. 2(pg.  295- 332) Google Scholar CrossRef Search ADS   18 Gérardin P..  Sur les représentations du groupe linéaire général sur un corps p-adique,  Séminaire Delange-Pisot-Poitou ,  1973 Paris Secrétariat Mathématique(pg.  1- 24) Théorie des nombres 14 19 Grothendieck A..  Éléments de géométrie algébrique 4: Étude locale des schémas et des morphismes de schémas 4,  Publications Mathématiques de l'Institut des Hautes Études Scientifiques ,  1967, vol.  32  20 Hales T. C..  A simple definition of transfer factors for unramified groups,  Representation theory and harmonic analysis ,  1993 Providence, RI American Mathematical Society(pg.  109- 34) Contemporary Mathematics 145 21 Kazhdan D..  On lifting,  Lie Group Representations 2 ,  1984 Berlin Springer(pg.  209- 49) Lecture Notes in Mathematics 1041 22 Kim J.-L.,  Murnaghan F..  Character expansions and unrefined minimal K-types,  American Journal of Mathematics ,  2003, vol.  125 (pg.  1199- 234) Google Scholar CrossRef Search ADS   23 Kottwitz R. E..  Isocrystals with additional structure 2,  Compositio Mathematica ,  1997, vol.  109 (pg.  255- 339) Google Scholar CrossRef Search ADS   24 Kutzko P. C..  On the supercuspidal representations of ,  American Journal of Mathematics ,  1978, vol.  100 (pg.  43- 60) Google Scholar CrossRef Search ADS   25 Landvogt E.. ,  A compactification of the Bruhat–Tits building ,  1996 Berlin Springer Lecture Notes in Mathematics 1619 26 Lang S..  On quasi algebraic closure,  Annals of Mathematics ,  1952, vol.  55  no. 2(pg.  373- 90) Google Scholar CrossRef Search ADS   27 Moy A..  Displacement functions on the Bruhat–Tits building,  The Mathematical Legacy of Harish-Chandra ,  2000 Providence, RI American Mathematical Society(pg.  483- 99) Proceedings of Symposia in Pure Mathematics 68 28 Moy A.,  Prasad G..  Unrefined minimal K-types for p-adic groups,  Inventiones Mathematicae ,  1994, vol.  116 (pg.  393- 408) Google Scholar CrossRef Search ADS   29 Moy A.,  Prasad G..  Jacquet functors and unrefined minimal K-types,  Commentarii Mathematici Helvetici ,  1996, vol.  71 (pg.  98- 121) Google Scholar CrossRef Search ADS   30 Murnaghan F..  Characters of supercuspidal representations of SL(n),  Pacific Journal of Mathematics ,  1995, vol.  170 (pg.  217- 35) Google Scholar CrossRef Search ADS   31 Murnaghan F..  Local character expansions and Shalika germs for GL(n),  Mathematische Annalen ,  1996, vol.  304 (pg.  423- 55) Google Scholar CrossRef Search ADS   32 Prasad G.,  Yu J.-K..  On finite group actions on reductive groups and buildings,  Inventtiones Mathematicae ,  2002, vol.  147 (pg.  545- 60) Google Scholar CrossRef Search ADS   33 Rapoport M..  The reduction of the Shimura variety associated to a torus  Forthcoming 34 Roche A..  Types and Hecke algebras for principal series representations of split reductive p-adic groups,  Annales Scientifiques de l'École Normale Supérieure ,  1998, vol.  31  no. 4(pg.  361- 413) Google Scholar CrossRef Search ADS   35 Rousseau G..  Immeubles des groupes réductifs sur les corps locaux,  1977  PhD Thesis, University Paris XI 36 Sally P. J.Jr,  Shalika J. A..  Characters of the discrete series of representations of SL(2) over a local field,  Proceedings of the National Academy of Sciences of the United States of America ,  1968, vol.  61 (pg.  1231- 7) Google Scholar CrossRef Search ADS PubMed  37 Serre J.-P.. ,  Local Fields ,  1979 New York Springer Graduate Texts in Mathematics 67 38 Serre J.-P.. ,  Lie Algebras and Lie Groups ,  1992 Berlin Springer Lecture Notes in Mathematics 1500 39 Serre J.-P.. ,  Galois Cohomology ,  2002 Berlin Springer Springer Monographs in Mathematics 40 Shimizu H..  Some examples of new forms,  Journal of the Faculty of Science: University of Tokyo Section IA: Mathematics ,  1977, vol.  24 (pg.  97- 113) 41 Silberger A. J.. ,  PGL2 over the p-adics: Its representations, Spherical Functions, and Fourier Analysis ,  1970 Berlin Springer Lecture Notes in Mathematics 166 42 Spice L..  Supercuspidal characters of over a p-adic field, ℓ a prime,  American Journal of Mathematics ,  2005, vol.  127 (pg.  51- 100) Google Scholar CrossRef Search ADS   43 Spice L..  Topological Jordan decompositions,  Journal of Algebra ,  2008, vol.  319 (pg.  3141- 63) Google Scholar CrossRef Search ADS   44 Springer T. A.. ,  Linear Algebraic Groups ,  1998 Boston, MA Birkhäuser Boston Progress in Mathematics 9 45 Springer T. A.,  Steinberg R..  Conjugacy classes,  Seminar on Algebraic Groups and Related Finite Groups ,  1970 Berlin Springer(pg.  167- 266) Lecture Notes in Mathematics 131 46 Tits J..  Reductive groups over local fields,  Automorphic Forms, Representations, and L-Functions: Part 1 ,  1979 Providence, RI American Mathematical Society(pg.  29- 69) Proceedings of Symposia in Pure Mathematics 33 47 Weil A.. ,  Basic Number Theory ,  1973 2nd ed New York Springer 48 Yu J.-K..  Construction of tame supercuspidal representations,  Journal of the American Mathematical Society ,  2001, vol.  14 (pg.  579- 622) Google Scholar CrossRef Search ADS   49 Yu J.-K..  Smooth models associated to concave functions in Bruhat–Tits theory,  2002  preprint Advance Access 2002-20. http://www.ims.nus.edu.sg/publications-pp02.htm © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press

Baldwin, Elizabeth;Swinarski, David

doi: 10.1093/imrp/rpn004pmid: N/A

Abstract We construct the moduli spaces of stable maps, , via geometric invariant theory (GIT). This construction is only valid over , but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, ; this is valid over . In another paper by the first author, a small part of the argument is replaced, making the result valid in far greater generality. Our method follows the one used in the case n = 0 by Gieseker in [9], 1982, Lectures on Moduli of Curves to construct , though our proof that the semistable set is nonempty is entirely different. References 1 Alexeev V.,  Guy G. M..  Moduli of weighted stable maps and their gravitational descendants,  Journal of the Institute of Mathematics of Jussieu ,  2006  preprint math.AG/0607683 2 Baldwin E..  A geometric invariant theory construction of moduli spaces of stable maps,  2006  DPhil thesis, University of Oxford 3 Baldwin E..  A GIT construction of moduli spaces of stable maps in positive characteristic,  Journal of the London Mathematical Society ,  2008  Advance Access doi: 10.1112/jlms/jdn014 4 Baldwin E..  A GIT construction of as a quotient of a subscheme of   Manuscript in preparation 5 Bayer A.,  Manin Yu..  Stability conditions, wall-crossing and weighted Gromov–Witten invariants,  2006  preprint math.AG/0607580 6 Behrend K.,  Manin Yu..  Stacks of stable maps and Gromov–Witten invariants,  Duke Mathematical Journal ,  1996, vol.  85  1(pg.  1- 60) Google Scholar CrossRef Search ADS   7 Dolgachev I. V.,  Hu Yi.  Variation of geometric invariant theory quotients,  Publications Mathematiques Institut de Hautes Etudes Scientifiques ,  1998, vol.  87 (pg.  5- 56) Google Scholar CrossRef Search ADS   8 Fulton W.,  Pandharipande R..  Notes on stable maps and quantum cohomology,  Algebraic geometry—Santa Cruz 1995 ,  1997 Providence, RI American Mathematical Society(pg.  45- 96) Proceedings of Symposia in Pure Mathematics 62 9 Gieseker D.. ,  Lectures on Moduli of Curves ,  1982 Bombay, India Tata Institute of Fundamental Research Tata Institute of Fundamental Research Lectures on Mathematics and Physics 69 10 Grothendieck A..  Éléments de géométrie algébrique 3: Étude cohomologique des faisceaux cohérents 2,  Publications Mathematiques Institut de Hautes Etudes Scientifiques ,  1963, vol.  17 pg.  91  11 Harris J.,  Morrison I.. ,  Moduli of Curves ,  1998 New York Springer Graduate Texts in Mathematics 187 12 Hartshorne R.. ,  Algebraic Geometry ,  1977 New York Springer Graduate Texts in Mathematics 52 13 Hassett B..  Moduli spaces of weighted pointed stable curves,  Advances in Mathematics ,  2003, vol.  173  2(pg.  316- 52) Google Scholar CrossRef Search ADS   14 Hassett B.,  Hyeon D..  Log canonical models for the moduli space of curves: First divisorial contraction,  2006  preprint math.AG/0607477 15 Hyeon D.,  Lee Y..  Log minimal model program for the moduli space of stable curves of genus three,  2007  preprint math.AG/0703093 16 Kim B.,  Pandharipande R..  The connectedness of the moduli space of maps to homogeneous spaces,  Symplectic Geometry and Mirror Symmetry ,  2001 River Edge, NJ World Scientific(pg.  187- 201) 17 Knudsen F. F..  The projectivity of the moduli space of stable curves 3: The line bundles on , and a proof of the projectivity of in characteristic 0,  Mathematica Scandinavica ,  1983, vol.  52  2(pg.  200- 12) 18 Morrison I..  Projective stability of ruled surfaces,  Inventiones Mathematicae ,  1980, vol.  56  3(pg.  269- 304) Google Scholar CrossRef Search ADS   19 Mukai S.. ,  An Introduction to Invariants and Moduli ,  2003 Cambridge, United Kingdom Cambridge University Press Cambridge Studies in Advanced Mathematics 81 20 Mumford D.,  Fogarty J.,  Kirwan F.. ,  Geometric Invariant Theory ,  1994 3rd ed Berlin Springer Results in Mathematics and Related Areas 34 21 Mustata A.,  Mustata A..  Intermediate moduli spaces of maps,  Inventiones Mathematicae ,  2007, vol.  167  1(pg.  47- 90) Google Scholar CrossRef Search ADS   22 Newstead P. E.. ,  Introduction to Moduli Problems and Orbit Spaces ,  1978 Bombay, India Tata Institute of Fundamental Research Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51 23 Parker A..  An elementary GIT construction of the moduli space of stable maps,  2006  preprint math.AG/0604092 24 Schubert D..  A new compactification of the moduli space of curves,  Compositio Mathematica ,  1991, vol.  78  3(pg.  297- 313) 25 Seshadri C. S..  Quotient spaces modulo reductive algebraic groups,  Annals of Mathematics , vol.  95  2(pg.  511- 56)  errata, Annals of Mathematics 96, no. 2 (1972): 599 CrossRef Search ADS   26 Seshadri C. S..  Geometric reductivity over arbitrary base,  Advances in Mathematics ,  1977, vol.  26  3(pg.  225- 74) Google Scholar CrossRef Search ADS   27 Swinarski D..  GIT stability for weighted pointed curves,  2008  preprint arXiv 0801.1288 28 Swinarski D..  Geometric invariant theory and moduli spaces of maps,  2003  Master's thesis, University of Oxford 29 Thaddeus M..  Geometric invariant theory and flips,  Journal of the American Mathematical Society ,  1996, vol.  9  3(pg.  691- 723) Google Scholar CrossRef Search ADS   30 Vakil R..  The moduli space of curves and its tautological ring,  Notices of the American Mathematical Society ,  2003, vol.  50  6(pg.  647- 58) 31 Vakil R..  Murphy's law in algebraic geometry: Badly-behaved deformation spaces,  Inventiones Mathematicae ,  2006, vol.  164  3(pg.  569- 90) Google Scholar CrossRef Search ADS   © The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. Oxford University Press