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204 IVAN SMITH genus from the above construction (choosing the braid to obtain the trivial sphere bundle Σ2×S 2 and not some twisted bundle). The symplectic surface represents a homology class 2m[Fibre] for some m and the unique complex representative of this fibre is again disconnected. The result of the fibre summation remains simply connected since in K3♯2CP 2 the fundamental group of the complement of a fibre is trivial. Now for any complex projective surface Z we may find a Lefschetz pencil of curves of some large genus. Blowing up the base-points of this pencil we have a Lefschetz fibration, inside which we can identify a trivial fibration Σr × D for a complex disc D ⊂ P 1 . Iterating the fibre sum construction above, we can find a symplectic curve of any large odd genus inside Σr × S 2 which is disjoint from some fibre S 2 and hence can be assumed to lie in the disc bundle Σr ×D ⊂ Z. The usual arguments show this is not the complex representative of the relevant homology class 2m[Fibre] (which is unique when the surface is 1-connected). This completes the proof. Remark 2.6. In complex geometry one is often as interested in nodal and cuspidal curves as smooth curves; for instance these appear as branch loci of generic projections of projective surfaces to CP 2 . Following Auroux’s work on symplectic four-manifolds [1] the symplectic geometry of these surfaces is now of interest. Moishezon [7] gave examples of nodal cuspidal curves in CP 2 which were not isotopic to complex nodal cuspidal curves; all of his examples contained cusps. The surfaces we construct in T 2 ×S 2 can be pushed down to S 2 × S 2 where they acquire nodes (but not worse singularities). <strong>For</strong> careful braids (for instance of several components) the resulting surfaces can be made symplectic. It is not clear if this is of interest, however, since the symplectic surfaces built in this way appear always to have nodes of both positive and negative intersection. Acknowledgements. The examples of the first section appeared in my D.Phil thesis and I am grateful to Simon Donaldson for general inspiration. Minor errors were kindly pointed out by Bernd Siebert and Ron Fintushel before I prepared the later drafts of the work, and it is a pleasure to acknowledge their contribution here. References [1] D. Auroux, Symplectic four-manifolds as branched coverings over CP 2 , Invent. Math., 139 (2000), 551-602. [2] J. Birman, Braids, Links and Mapping Class Groups, Princeton University Press, 1975. [3] K.S. Brown, Cohomology of Groups, Springer, 1982. [4] R. Fintushel and R. Stern, Symplectic surfaces in a fixed homology class, J. Diff. Geom., 52 (1999), 203-222.
SYMPLECTIC SUBMANIFOLDS FROM SURFACE FIBRATIONS 205 [5] H. Geiges, Symplectic structures on T 2 bundles over T 2 , Duke Math. J., 67 (1992), 539-55. [6] R. Gompf, A new construction of symplectic manifolds, Annals of Math., 142 (1995), 527-595. [7] B.G. Moishezon, Stable branch curves and braid monodromies, Algebraic Geometry, Chicago 1980, LNM, 862, Springer, (1981), 107-192. [8] W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-8. Received May 11, 1999 and revised October 5, 1999. New College Oxford OX1 3BN England E-mail address: smithi@maths.ox.ac.uk
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Pacific Journal of Mathematics Volu
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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EIGENVALUES ASYMPTOTICS 3 (i) If pm
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EIGENVALUES ASYMPTOTICS 5 Corollary
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Thus, by the Itô formula, we have
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Therefore, K1(t) ≡ t 2−d/2 ≤
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EIGENVALUES ASYMPTOTICS 11 The chan
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EIGENVALUES ASYMPTOTICS 13 Here Res
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IMAGINARY QUADRATIC FIELDS 17 Table
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IMAGINARY QUADRATIC FIELDS 19 and t
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IMAGINARY QUADRATIC FIELDS 21 Propo
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〈σ 〈σ〉 〈σ, τ〉 2 〈1
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IMAGINARY QUADRATIC FIELDS 25 Then
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IMAGINARY QUADRATIC FIELDS 27 were
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IMAGINARY QUADRATIC FIELDS 29 Thus
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IMAGINARY QUADRATIC FIELDS 31 [3] ,
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34 CARINA BOYALLIAN spherical serie
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36 CARINA BOYALLIAN Here I G MAN (
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38 CARINA BOYALLIAN where k ≤ [ n
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40 CARINA BOYALLIAN The case G = Sp
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42 CARINA BOYALLIAN we can, since i
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44 CARINA BOYALLIAN and this formul
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46 CARINA BOYALLIAN Here, J(ν) int
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48 CARINA BOYALLIAN [V] D. Vogan, R
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50 ROBERT F. BROWN AND HELGA SCHIRM
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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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98 GILLES CARRON Proposition 2.7. N
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100 GILLES CARRON l’espace des 1
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 113 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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BRAIDED OPERATOR COMMUTATORS 119 Mo
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BRAIDED OPERATOR COMMUTATORS 121 It
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E-mail address: prosk@imath.kiev.ua
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124 DANIEL J. MADDEN times. Further
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126 DANIEL J. MADDEN Further, s t
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128 DANIEL J. MADDEN Now the whole
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130 DANIEL J. MADDEN and β = 1 −
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132 DANIEL J. MADDEN We can simplif
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134 DANIEL J. MADDEN surd, but we n
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136 DANIEL J. MADDEN and d = d(u, v
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138 DANIEL J. MADDEN (6l + 7) k +
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140 DANIEL J. MADDEN Then the conti
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142 DANIEL J. MADDEN Thus n + 1 1 =
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144 DANIEL J. MADDEN If we take b =
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146 DANIEL J. MADDEN b, 2b(2bn + 1)
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TANGLES, 2-HANDLE ADDITIONS AND DEH
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Page 212 and 213: 208 PAULO TIRAO Theorem. The affine
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HÖLDER REGULARITY FOR ∂ 255 Refe
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