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Index of Notation

# #/

A(M)

Ao(M)

A(M, k)

A(P)

A(P) Aoo(P)

Aoo(P) AO,l(V)

AO,l(V) A~l(V)

A~(V)

adP adV

B

B(g, d)

Bn.c.j(g, d)

connected sum of manifolds 12 fiber connected sum of Coo elliptic surfaces 161

linear equivalence of divisors 11

automorphisms of H2(M) preserving the quadratic form qM

of M 12 automorphisms of the quadratic form of M which are the iden­tity on all classes coming from the boundary 185 automorphisms of the quadratic form of M preserving k 393

space of all irreducible L~ connections on the principal bundle P 231 space of all L~ connections on the principal bundle P 230 space of all irreducible Coo connections on the principal bundle P 230 space of all Coo connections on the principal bundle P 230 space of all simple L~ (0, I)-connections on the vector bundle V 293 space of all L~ (0, I)-connections on the vector bundle V 293 space of all simple Coo (0, I)-connections on the vector bundle V 283 space of all Coo (0, I)-connections on the vector bundle V 283

adjoint bundle associated to the principal bundle P 230 vector bundle of trace zero endomorphisms of the vector bun­dle V 300

the set of points a which satisfy 184 Ixl2 dJ.La ~ f 250 the automorphisms of the elliptic surface B preserving the section E 76

dimension of a maximal positive definite subspace of H 2 (M ; lR) 12 dimension of a maximal negative definite subspace of H 2 (M; lR) 12

sheaf of holomorphic cross-sections of the elliptic surface B 77

the set of isomorphism classes of elliptic surfaces S with a section with Euler number 12d over a base of genus 9 63 the set of isomorphism classes of elliptic surfaces in B(g, d) with nonconstant j-invariant 63

Bg.b.(g, d)

Bnod.(g, d)

BrB

BE

C={Ctlt;::::l}

C(V) C±(V)

x(-rr,5U(2»

fJ fJA 'VA

Det(>.) detAs(D) detBGs(D) detR7r. Div, div

L1T

d(c)

Diff+M D(M) Do(M)

D*(M)

D(M,k) d(fr E)

degx <p

<E -Es

F(J, G)

Index of Notation 507

the set of all surfaces 5 in B(g, d) such that (5, js) has generic branching behavior 64 the set of all surfaces 5 in B(g, d) such that 5 is nodal 64

the Brauer group of B 86

UE1<EZ(KEI,EEI) 261

group of conformal contractions of 54 251

Clifford algebra of V 374 elements of even or odd degree in C(V) 375

character variety of representations of 7r in 5U(2) 366

the Dirac operator 235 the Dirac operator coupled to the connection A 235 differential operator associated to the connection A 231

determinant line bundle associated to E+ (A) 238 the Atiyah-Singer determinant line bundle 385 the Bismut-Gillet-Soule determinant line bundle 391 the holomorphic determinant line bundle 383 the zero set of the natural section of the inverse of the deter­minant line bundle 384

the pullback of Div(fJT) to XO(P) 235 the image of D~ in XO(P) 235 the subset of Xo(P,g) such that either the background con­nection lies in DT or the singular support meets T 268 smooth divisor representing p,([T]) and lying in an arbitrarily small neighborhood of DT(P, g) 268 one-half the expected dimension of the moduli space, namely 4c - 3(1 + bt)/2 226

group of orientation-preserving self-diffeomorphisms of M 12 the subgroup of A(M) induced by Diff+ M 12 the subgroup of D(M) induced by compactly supported orien­tation-preserving self-diffeomorphisms 185 the subgroup of D(M) of elements of real spinor norm one 393 the subgroup of D(M) of elements fixing k 393, 405 the largest dimension of a stratum 17' with 17' < E 261

local degree at x of the map <p whose differential at x is Fred­holm of index zero 350, 351

universal elliptic curve over S) 37 the intersection form corresponding to the negative of the ma­trix for the root system Es 180 the intersection form corresponding to the negative of the ma­trix for the extended root system Es 181, 182

the Atiyah-Hitchin-Singer deformation complex for the ASD connection A 237

the set of elliptic surfaces without multiple fibers with the invariants J and G 76

508 Index of Notation

Fa

g(P) g(P)o

g~

gC gC,O

G(nl, ... ,nt)

Gs

7c(M, (3) 7~t(M, (3) 7c(M, (3) 7c,I(M)

r(V)

Gk ril

fj fjo

H2(M;7!..) 1tl(adP), 1t~(adP) 1to,Q(ad V), 1t0 ,Q (End V)

10 II" I(N)

j j

js J J(C) J(S)

the curvature of the connection A 232 the self-dual part of FA 255 the principal SO(4) bundle induced by the tangent bundle of M 251 the restriction of the fiber product over M (PI, g) x (M x ... x M - .1) 258 union of the strata of X6(P, g) whose points have trivial back­ground connections 263

the L~-gauge group of the bundle P 231 the based gauge group of the bundle P 233 the complex gauge group of C'~o bundle isomorphisms of the vector bundle V 285 the L~ complex gauge group 293

the L~ based complex gauge group 330

the semi-direct product

(TI!=l SO(4)) x ((TI!=l SU(2)) /{±1}) ~6(nl' ... ,nt) 252

the homological invariant of the elliptic surface S 43

Donaldson polynomial for M 226, 277 stable Donaldson polynomial for M 271, 273 restricted Donaldson polynomial for M 278 generalized Donaldson polynomial for M 411, 412

group of units in C+ (V) of norm one 375

the elements in SO(V; q fixing k E V 397 group generated by the reflections in the set .1 185, 401

the upper half plane 36 fj - P SL(2, 7!..) . {O, 1728} 40

H2(M; 7!..)/Torsion 12 harmonic forms for the elliptic complex E+ (A) 324

harmonic forms for the elliptic complex ({}o,e(M; ad V), [}) or ({}o,e(M; End V), [}) 294

a Kodaira fiber which is a cycle of n rational curves 51, 52 a multiple fiber whose reduction is a cycle of n rational curves 35 a Kodaira fiber 60 a Kodaira fiber 181

generalized connections whose singular support meets N 263

the classical j-function on the upper half plane 37 the function induced by j on the quotient of the upper half plane by PSL(2, 7!..) 37 the j-function of the elliptic surface S 43 1728js 43 the Jacobian of the algebraic curve C 345 the Jacobian (or basic) elliptic surface associated to the elliptic surface S 46

JS

Kx /1,(8)

K(V)

).c,N,I(M,13)

p,(a)

M(P,g)

M(P,g)

M~P,g) M (P,g) M(P, >.)

[M(P,g)]6

!mh01(V)

vltho1(V)

!mc(8, L)

!m~(8,L) !msch

!m(C) !mo(C) !m(C)

!mo(C)

MM,V

Index of Notation 509

the subsheaf of B consisting of holomorphic sections of 1r pass­ing through the identity components of the reducible fibers 82 the sheaf Rl1r.Os/Rl1r;l!..s on the elliptic surface 8 85

the canonical bundle of the manifold X 11 the Kodaira dimension of 8 19

the Kuranishi model for the space of deformations of the com­plex structure on V 295

the operator ('V:4, 'V1) 238

coefficients of the dual element to p in the expansion of 'Yc(M,13) 278 generalized polynomial invariants arising from the connected sum with the negative definite manifold N 426

the conjugate complex manifold corresponding to M 206

the map H4-i(Mj Z) ----+ Hi(X(P)j Z) defined by slant prod­uct -Hpl/a) 233 80(3)-equivariant class in Hi(XO(PIT)), where T is an em­bedded 2-manifold 235 extended p,-map 272 extended p,-map defined on the stable elements 271 normalized degree or slope of the holomorphic bundle V with respect to a fixed Kahler class or ample line bundle 322

the moduli space of all irreducible g-ASD connections on P modulo gauge equivalence 236 the moduli space of all g-ASD connections on P modulo gauge equivalence 236 the moduli space of all g-ASD connections on P 236 the based moduli space of all g-ASD connections on P 237 the parametrized moduli space of all gt-ASD connections on P where>. is the path {gt} 239 a-approximation to the fundamental class of M(P, g) in X6(P,g) 260

the set of simple holomorphic structures on V 294

the set of all holomorphic structures on V (usually assumed to have trivial determinant) 285

the set of holomorphic structures on V with trivial determi­nant 301 the moduli space of L-stable rank two holomorphic bundles with Cl = 0 and C2 = C 328 the based moduli space corresponding to !mc(8, L) 330 scheme version of !mc,S,L 337 moduli space of stable bundles of degree zero on C 356 subset of !m( C) of bundles with trivial determinant 356 compact moduli space of equivalence classes of semistable bun­dles of degree zero on C 357 subset of !m(C) of bundles with trivial determinant 357

global moduli functor of holomorphic structures on V 303

510 Index of Notation

Mv Mc,S,L

MO c,S,L

M sch

M,M

Mo,Mo

° =-=<> Mo,Mo

mw

N6 (p,g)

N6(P,g)

D(P), D'(P)

Pn(8) pg(8)

Pic X PicdC

qM q(8)

1?,(7r, G)

1?,(N)

local moduli functor of germs of deformations of V 304 global moduli functor of L-stable bundles on 8 with Cl = 0 and C2 = C 334 based version of Mc,S,L 336 scheme version of Mc,S,L 337 deformation functors for stable or semistable bundles on C 356, 357 deformation functors for stable or semistable bundles on C with trivial determinant 357 rigidified versions of Mo, Mo 363

multiplicity of the irreducible component W in the complex space Z 349

almost ASD connections on P which are almost flat away from certain points of concentrated curvature, and which lie in the span of the eigenvectors with small eigenvalues for the opera­tor \71 0 (\71)* 255 the image of N6(P,g) in X(P) 256

sets of generic metrics 237, 249

the nth plurigenus of 8 18 the geometric genus of 8 18

the Picard group of X 11 the set of isomorphism classes of line bundles on C of degree d 11

the orbifold fundamental group of C 145

the 8U(2)-bundle over M with C2 = C 226 first Pontrjagin class of the universal 80(3) bundle over X(P) x M 232 the image of 1 under the J1-map 269 the restriction of the extension of p to a stratum E 270

projection onto the space of self-dual 2-forms 242

intersection form of the oriented 4-manifold M 12 the irregularity hO,1(8) 19

restriction of connections to the 2-manifold T 234 restriction of connections for the quotient space of connections modulo the based gauge group 234

variety of representations of 7r in the Lie group G 366

reflection about 0 185

reducible ASD connections on the negative definite manifold N 431

the minimal model of the complex surface 8 21 the set of all elliptic surfaces without multiple fibers whose Jacobian surface is B 75 the set of all algebraic surfaces in man (B) 86

s

§±

Spin(V)

6 n Si(M) 6(nl, ... ,nt)

Sym*V S*V

SO(V, q), SO (V)

Index of Notation 511

all complex surfaces S such that K,(S) ~ 0 or K,(S) = -00

and S is algebraic or is deformation equivalent to a (possibly blown up) Hopf surface 221

the bundles of plus and minus spinors 235, 376 the spin group associated to the quadratic form on V 375

the symmetric group on n letters 68 the (!th symmetric product of M 244 the largest subgroup of the permutation group of the factors preserving the multiplicities 252

the top stratum M(P,g) of X(P,g) 261 the real spinor norm 397

the symmetric algebra on V 395 the invariants of the symmetric group acting on the tensor algebra 395 the group of linear maps of V preserving the quadratic form q 397

SO(V, q; k), SO(V, k) the subgroup of SO(V, q) fixing k 397

T(S,{td'{~i})

T(S)

e {)

Vo(P,g) Vo(P, oX)

W(C,P) Wo,p(P,g)

W3,p(P,g) W>.(P,gM) Wf(P,gM)

x Xoo(P)

Xoo(P) X(P)

X(P)

XO(P), XO(P) X(P,g) X(P,>..) Xo(P,g)

Y(P,g)

yO(P,g)

set of elliptic surfaces locally isomorphic to S except over the ti and with the same basic elliptic surface 102 set of elliptic surfaces locally isomorphic to S and with the same basic elliptic surface 102

the theta divisor of a Jacobian 345 a theta function with characteristic 345

thickened moduli space 247, 257 parametrized version of V.(P,g) 260

subset of A(P) with bounds on curvature 239, 240 subset of Vo(P, g) whose measures do not have singular sup­port at p 257 based version of Wo,p(P,g) 257 generalized connections with small curvature in a ball B>. 434 based version of W>.(P,gM) 435

universal cover of neighborhood of a Ik fiber 54

space of irreducible Coo connections on P modulo gauge equiv­alence 230 space of all connections on P modulo gauge equivalence 230 space of irreducible L~ connections on P modulo gauge equiv­alence 232 space of all L~ connections on P modulo gauge equivalence 231 based versions of X(P) andX(P) 233 Uhlenbeck compactification of the moduli space 244 parametrized version of X(P,g) 245 thickened completion of the moduli space 260

set of points of X(P,g) whose associated measure does not have support at p 246 a based version of Y(P, g) 246

512 Index of Notation

Y(P, ..\), Y°(p,..\) ~(P,N,g)

YoO(P,N,g)

Z(Q,E)

ZO(Q, E) ZO((nl,'" nt), E) Z(E, E)

Ze(C,(})

Z~(C, (})

parametrized versions of Y(P,g) and yO(P,g) 246 set of points of X 0 (P, g) whose associated measures do not have support on N 263 based version of Yo (P, N, g) 263

the intersection of A(Q, E) with the set of those points a E X(Q,go) whose center of mass is the north pole 250 based version of Z ( Q, E) 251 I1~=1 ZO (Qnp E) 253

- ° ° Fr(E) XC(nl, ... ,nt) Z ((nl, ... ,nt},E) 253

the divisor on me (S, L) corresponding to the smooth curve C C S and the theta characteristic () on C 344, 345 the divisor on m~ (S, L) corresponding to Ze (C, ()) 344

Index

adjunction formula 194, 418, 454, 464 algebraic elliptic surface see elliptic

surface, algebraic almost complex structure 282, 284,

308, 312~313, 314 analytic function on a Banach space

285~293

anti-self dual (ASD) see connection, anti-self-dual

Artin ring 95, 337, 350 Artin, M. 32 ASD connections see connection, anti­

self-dual ASD Yang-Mills equations see Yang­

Mills equations Atiyah's theorem on vector bundles

over elliptic curves 447, 448, 452, 456

Atiyah, M. 137 Atiyah-Hitchin-Singer deformation

complex 237, 319 Atiyah-Singer determinant see deter­

minant line bundle, Atiyah-Singer Atiyah-Singer index theorem see index

theorem, Atiyah-Singer Averbuh, B. G. 32

background connection 228, 229, 244~247, 249~251, 253~255, 257~261,

263, 265, 266, 430, 434~436 Barlow surface see surface, Barlow base change 33, 38, 47, 53, 95, 98,

99, 115, 116, 163, 164, 338, 382, 452, 462, 465, 466, 482, 483, 488

based gauge group see gauge group, based

based moduli functor see moduli functor, based

based moduli space see moduli space, based

basic elliptic surface see elliptic surface, basic

basic member 76, 85, 93, 205

basic surface see elliptic surface, basic Bauer, S. 223, 497 Bianchi identity 284, 294 big diffeomorphism group

see diffeomorphism group, big Bismut-Freed determinant 374 Bismut-Gillet-Soule determinant 374,

389~392

block 71~75

blowing down 17, 21, 24~27 blowing down in families 21, 24~27,

35 blowing up 17, 19, 131, 132, 163 blowup 4, 5, 10, 17~20, 23, 27, 28, 36,

51, 59, 131, 132, 137, 154~ 157, 163, 164, 181, 221~225, 394, 411~416, 418, 426, 491~498

Bogomolov's inequality 367 Bogomolov, F. 366 Bogomolov-Miyaoka-Yau inequality

30, 494, 495 Bombieri's theorem on the canonical

map 33 Bombieri, E. 33 Borel measure 240 botany 23 Bott, R. 137 bounded homogeneous polynomial

285 Brauer group 86, 109, 335 bridge 72~ 75 Brieskorn, E. 33, 115 Brussee, R. 499

c-generic 339 c-stable 271, 276 c-suitable 446~448, 450, 451, 453,

454, 468, 476, 477 436, 438 Coo -elliptic surface see elliptic surface,

Coo COO-isomorphism of elliptic surfaces see

elliptic surface, Coo -isomorphism of

514 Index

canonical bundle 2, 10, 11, 16-20, 24, 25, 130, 160, 341, 343, 376, 394, 406, 410, 418, 450, 477

canonical bundle formula for an elliptic surface 16, 36, 49, 50, 129, 130, 454, 464, 475

Cartan matrix 179, 187 Cartan's privileged neighborhood

theorem 298 Case (A) 447, 448, 451, 453-455, 469,

479, 480, 485, 497 Case (B) 448, 453-456, 480, 481 Castelnuovo's theorem 2, 25 Castelnuovo-deFranchis theorem 20,

30, 494 Castelnuovo-Enriques theorem 20 categorical quotient 362, 364 Christoffel symbol 378 Clifford algebra 374, 375 Clifford multiplication 375-377 Clifford's theorem 360 closure of modules theorem 288, 289 coarse moduli space see moduli space,

coarse compatible lift 41-47, 67 completion of the moduli space

239-246 thickened 247, 260--263

complex gauge group see gauge group, complex

complex space 11 defined by a Fredholm map

290--293 defined by a Fredholm section 292,

293 defined by a Fredholm section 265,

complex torus 22, 30--33, 92, 131, 154, 155, 199, 209, 214, 222, 22~ 444, 458, 494, 495

cone bundle 228, 247, 251, 253, 255, 258-260, 265, 271

cone structure 248, 251, 253, 258 conformal length 432 conjugate complex manifold 5, 139,

206,409 connected sum 2, 9, 10, 12, 161, 229,

271, 356, 394, 411-414, 416-419, 425, 426, 432, 433, 435

connection (0,1)- 283-285, 294, 300, 301, 307,

314, 324, 327, 331 integrable 284, 285 background 228, 229, 244-247,

249-251, 253-255, 257-261, 263, 265, 266, 430, 434-436

compatible with the complex struc-ture 280--283, 313, 320, 323, 333

Ehresmann 280, 284, 314 fiat 88, 249, 251, 366, 369, 379 Hermite-Einstein 323 Hermitian 281, 323 irreducible 226, 227, 230, 231, 234,

236, 237, 249, 256, 271, 279, 324-327, 330, 347, 365, 429, 431, 438

Levi-Civita 376 product 240, 243, 249, 256, 258 reducible 231, 246, 249, 250, 257,

274, 338, 348, 366, 429-432, 436, 438-440

trivial 228, 238, 243, 246, 247, 255-260, 348, 370, 373, 430, 431, 433,435

connections, space of gauge equivalence classes 226, 230-234 based, space of gauge equivalence

classes 233, 431 contains in the sense of complex spaces

288 curvature 230, 232, 282, 283

(0,2)- 284, 285, 293 concentrated 228, 254

cyclic monodromy 195-197, 201, 202, 205, 209, 210, 214, 215

D-approximation to the fundamental class 260--262

Dabrowski, K. 133 deformation, unobstructed 301 deformation equivalence 4, 5, 10,

14-22, 25, 28, 29, 31-34, 57, 61, 81, 93, 103, 138-140, 157, 201, 221-224, 409, 442, 445, 497

deformation equivalence through ellip­tic surfaces 35, 57, 64-67, 79, 81, 82, 103-107, 110--132, 158, 159, 201, 205-209, 211, 215

deformation invariance of Kodaira dimension 20, 26, 27 of the plurigenera 20, 24, 26, 27,

128-130 deformation type 4-7,16, 17,28,195,

497 degree, local, for a Fredholm map of

index zero 352, 355 Dehn twist 167 Dehn-Nielsen theorem 146-154 Deligne, P. 401 Deligne-Mumford theorem 125 descent conditions 473

determinant line bundle 238, 340, 352, 355, 374 Atiyah-Singer 235, 374, 385-389,

391 Bismut-Freed 374 Bismut-Gillet-Soule 374, 389-392 holomorphic 341, 342, 344, 364,

366, 374, 379-385, 387-389, 391 diffeomorphism group, big 9, 393,

394, 405-409, 413, 415, 416, 418, 420, 425, 427, 490, 499 Donaldson polynomial

of a 4-manifold with 9, 393, 406-408, 490, 499

Dirac operator 235, 268, 269, 274, 340, 341, 346-348, 374-379

directional derivative 420 distribution

horizontal 232, 280, 282, 331 integrable 308 involutive 308

Dolgachev surface see surface, Dolgachev

Dolgachev, I. 16, 75, 86 Donaldson polynomial invariant 4,

226, 227 full 229, 230, 276, 278 generalized 411-413 restricted 278 stable 229, 271-273, 276

Donaldson's theorem on connected sums 425 on definite 4-manifolds 3, 425 on diffeomorphisms of a K3 surface

194,408 on stable bundles and ASD

connections 4, 279, 323, 324 on the dimension of the moduli

space 339, 372 on the failure of the h-cobordism

theorem 3,4 on the nonvanishing of the poly­

nomial invariants of an algebraic surface 4, 279, 340, 356-373, 409

on the orientation of the moduli space 329

on the polynomial invariant of a blowup 411

Douady, A. 297, 308 Dynkin diagram 179, 181, 183, 184,

187-192, 401, 403

Ebeling's theorem 8, 191, 192, 194, 405

Ebeling, W. 403, 410

Index 515

Ehresmann connection see connection, Ehresmann

Eichler-Siegel transformations 399, 404, 405

elementary equivalence 172, 175-179 elementary transformation of a family

of vanishing arcs 168, 169, 171 elementary transformation of a ruled

surface 28 elliptic fibration 7, 23 elliptic regularity 237, 242, 256 elliptic structure 23 elliptic surface 4, 7, 23, 34-68, 75-132

algebraic 5, 6, 86, 87, 106, 107, 156 basic 16, 45-48, 56, 75, 82, 85, 89,

93, 95, 96, 98, 102-109, 113, 123-125, 127, 142, 159, 196, 202, 203,442

C oo_ 7, 138, 139 C oo _, basic 196, 202, 212 COO-isomorphism of 138-140 deformation equivalence of

35, 110-134 diffeomorphism classification of 5 Donaldson polynomials of 476, 491,

497 family of 35 isomorphism of 34 nodal 63, 141 properly 23 rational 51, 80, 131, 132, 164,

181-189, 420, 493 relatively minimal 23, 34, 51, 56 ruled 51, 131, 132, 199, 209, 217,

222, 224, 420, 492-495 with a section 34, 56-68 with Euler number zero 123-127,

154 with multiple fibers 35, 48-51,

95-110 with positive Euler number 122,

123 without multiple fibers 75-95

Enriques surface see surface, Enriques Enriques-Kodaira classification

2, 14-27 equivalence of families of bundles 303 equivalence of quadruples 126, 202,

206-208 equivalence, oriented, of quadruples

126, 127, 202, 206-208 Euler class 7, 89, 124 Euler number of an orbifold 146 exceptional classes 411-416, 419, 421,

425, 426, 428

516 Index

exceptional curve 16, 17, 21, 23, 24, 28, 154, 155, 394, 491, 493, 494, 497 stability of 24

expected dimension 302, 325, 340, 349,461

extended G-equivalence 361-363 extension of the J.t-map see J.t-map,

extension of

family of elliptic surfaces 35 of surfaces 17 of vector bundles 303

fiber connected sum 138, 161, 162, 164, 165, 170, 189, 190, 195, 493, 497

fine moduli space see moduli space, fine

fiat connection see connection, fiat orbifold see orbifold, fiat

fiat base change 462, 465, 466, 482, 483,488

foliation, complex 308-311 formal orientation 330, 355 fractional linear transformation 36 Fredholm 235, 290--292, 294-296, 328,

352, 355, 385 Freed-Uhlenbeck theorem see generic

metrics theorem Freedman, M. 3, 31, 161, 410, 496 Frobenius theorem 308, 312, 313 functional (j)-invariant 43 functional invariant 76 fundamental group of an elliptic

surface 157-159, 198-201

G-linearization 362, 365 gauge group 3, 231

based 233, 330 complex 280, 285, 300

gauge transformation, group of see gauge group

general type, surface of see surface of general type

generalized Donaldson polynomial invariant see Donaldson polynomial invariant, generalized

generic branching behavior 63-68 generic metrics theorem 237, 280, 339 geography 23 geometric genus 18 geometric invariant theory 334,

337-339, 357, 359, 362

geometric representatives for the J.t-map 234, 235, 268-270, 340, 341, 374, 428-431, 433

geometrically ruled surface see surface, geometrically ruled

Gieseker compactification 339 Gieseker, D. 334, 337-339, 357, 364,

373 GIT (geometric invariant theory) 363 global moduli functor see moduli

functor, global Griffiths, P. 282 Grothendieck, A. 336, 338, 374 Grothendieck-Riemann-Roch theorem

342, 343, 488

h-cobordism theorem 1, 3, 4, 137 Hambleton, I. 136, 137 Harvey, R. 201 Hermite-Einstein connection

see connection, Hermite-Einstein Hermitian connection see connection,

Hermitian Hermitian holomorphic vector bundle

281,323 Hilbert scheme 33 Hirzebruch index theorem see index

theorem, Hirzebruch Hodge *-operator 12, 236, 294, 319 Hodge decomposition 319 Hodge index theorem see index

theorem, Hodge Hodge metric 9, 323, 324, 338, 339,

341, 346, 352, 353, 369, 379 Hodge structure 42, 86, 87, 106 Hodge theory 2, 6, 86, 87, 91, 92,

294,323 homogeneous polynomial, bounded, on

a Banach space see bounded homo­geneous polynomial

homological invariant 16, 43, 45, 61, 76, 88, 93, 210

Hopf surface see surface, Hopf Hopf surface, primary see surface, Hopf Hopf surface, secondary see surface,

Hopf horizontal distribution see distribution,

horizontal horizontal lift of a vector field 282,

332 hyperbolic orbifold see orbifold, hyper­

bolic hyperelliptic surface see surface, hyper­

elliptic

!itaka, S. 20, 21, 224 index theorem

Atiyah-Singer 237, 343 Hirzebruch 30 Hodge 19, 367, 373, 406, 446, 454

Inoue surface see surface, Inoue integrable (O,l)-connection

see connection, (0,1) integrable distribution see distribution,

integrable involutive distribution see distribution,

involutive irreducible connection see connection,

irreducible isomorphism of elliptic surfaces

see elliptic surface, isomorphism of Itoh, M. 332

j-invariant 16, 43, 44, 47, 56, 58, 60-68, 84, 113, 121 constant 56, 60-63, 84, 88, 92, 157

Jacobian surface J(8) 46, 47, 75, 76, 86,100,114,442,443,447,453

Jacobian variety 130, 345, 444, 458

K3 surface see surface, K3 Kahler identities 325, 326 Kato's inequality 243 Kneser, M. 192, 194 Knudsen-Mumford determinant 374 Kobayashi, S. 323 Kodaira, K. 2, 3, 7, 14, 20, 21, 24,

32-34, 43, 46, 52, 57, 60, 75, 76, 81-84, 86, 93-95, 101, 103, 111, 129, 132-134, 181, 221, 495

Kodaira dimension 4, 7, 14-16, 19-31, 50, 51, 128, 130-132, 155, 156, 221, 223-225, 394, 494, 498

Kodaira surface see surface, Kodaira Kodaira-Spencer complex 293, 324 Kodaira-Spencer theory 6, 9,

279-280, 390 Koszul-Malgrange theorem 282 Kotschick, D. 498 Kreck, M. 497 Kronheimer, P. 497,498 Kuranishi map 295, 297-299, 301,

305, 327, 328, 355 Kuranishi model 293-303, 305, 314,

319, 327-329, 335-337, 355 Kuranishi obstruction map

see Kuranishi map Kuranishi's theorem 9, 279, 305-318,

331 Kuranishi, M. 279, 280

Index 517

Lawson, B. 201 leading coefficient 442, 444, 476, 491,

497 lens space 133, 136, 137 Leray spectral sequence 48, 79, 80,

83, 86, 87, 89, 90, 93, 110, 121, 122, 201, 265, 449, 463, 464, 473, 474, 482,483

Levi-Civita connection see connection, Levi- Civita

Li, J. 339 Libgober, A. 410 Livne's theorem 8, 139, 172-179 local complete intersection 350 local complete intersection, moduli

space is a 302 local degree see degree, local local moduli functor see moduli

functor, local local moduli space see moduli space,

local local monodromy 44, 45 locally semi universal 305 logarithmic transform 7, 16, 91,

95-113, 122-127, 165, 170, 200-204, 442, 493 coo_ 138, 143-145, 196, 197,

202-204, 210, 212, 213 long vector 180, 188 Liibke, M. 33, 323, 495

J,t-map 226-229, 233-235, 340-345, 370, 371, 374, 428-431, 433, 444, 461, 477-479, 486, 487, 490 extension of 228, 263-273

Maier, F. 33, 495 manifold point 49 Maruyama, M. 338 Matsumoto, Y. 164, 172 Maurer-Cartan form 280 Milnor number of a cusp 183 minimal 4-manifold 419 minimal complex surface 17-20, 22,

23, 25-27, 31-33, 155, 394 minimal model

of a 4-manifold 10, 415-420, 492 of a complex surface 16, 21, 22, 24,

27, 156, 420, 492 minimal, strongly 419, 420, 492, 494,

499 miniversal 305 Miyaoka, Y. 201 moduli functor 342-344, 357, 364

based 336, 337 global 280, 303, 304, 334-336

518 Index

moduli functor local 9, 279, 280, 304, 305, 337 rigidified 363, 364

moduli space based 237, 246, 330, 331, 336, 337,

342, 344, 379 coarse 57, 63, 280, 335, 337, 338,

357, 364, 368 fine 335 local 279, 305

Moishezon, B. 33, 56, 64, 66, 111, 113, 122, 138, 158, 164, 169, 172, 196, 409, 410

monodromy group 18 Mordell-Wei! group 46, 109 morphism from a complex space to a

Banach space 287, 288 Mrowka, T. 497, 498 multiple fiber 16, 35, 95-120, 125-127 multiple fiber with singular reduction

35, 96-100, 113-122 multiple points 49 multiplicity

of a multiple fiber 15, 35, 49, 50 of a point of an orbifold 15, 49 of an irreducible component of a

complex space 349, 350, 353, 355 multisection 34, 46, 444, 446, 454,

478 Mumford, D. 344, 357, 359, 382

Nakai-Moishezon criterion 446 Narasimhan-Seshadri theorem 324,

365,366 negative definite 4-manifold 10, 276,

356, 394, 425, 426, 428 Newlander-Nirenberg theorem 283 Newlander-Nirenberg theorem,

parametrized version 312 Newlander-Nirenberg theorem, real

analytic case 308, 311, 312 nodal see elliptic surface, nodal Noether's formula 2, 19, 30 normalized degree 321, 323

O'Grady, K. 491,499 obstruction bundle 247 obstruction space 302, 324, 355 Ogg, A. 106 Okonek, C. 33, 403, 495 one-parameter subgroup criterion 359 orbifold covering 145, 146, 149, 156,

198 orbifold fundamental group 107,

145-154, 158, 198, 493

orbifold map 146-154, 153 orbifold

bad 146, 198, 199, 210 base, of an elliptic surface 7, 15,

49, 51, 105, 111, 122, 125-127, 131, 132, 138, 146, 154-158, 198-202, 205, 207-220, 224, 409, 492,493

flat 51, 146, 154-156, 198, 210, 211, 214-216, 492, 493

good 146-154, 198 hyperbolic 51, 146, 154-156, 198,

210, 211, 213, 492-493 spherical 51, 128, 146, 198, 210,

216-220 orientation of the moduli space 227,

238, 260, 329, 330, 341, 354 orientation, formal 330, 355 oriented ordered basis 36-38

pair of pants 149, 150, 151 period matrix 345 permissible representation 68, 70 permissible sequence 69, 70 Picard-Lefschetz formula 55 Pidstrigach, V. 498 plurigenus 2, 4, 10, 11, 18, 24-27, 50,

128, 129, 498, 499 Poincare bundle 331-333 Poincare conjecture 1, 2 Poincare line bundle 449-452, 459,

460, 483, 487, 489 first Chern class of 489 relative 451, 452

polarization map 396 Pontrjagin square 233 primary Hopf surface see surface, Hopf properly elliptic see elliptic surface,

properly

Qin, Z. 31, 498 quadratic term of the Kuranishi map

299,300 Quillen, D. 374 Quot scheme 338

rational ruled surface see surface, rational ruled

rational surface see surface, rational real spinor norm see spinor norm, real reduced expression 172-174 reducible connection see connection,

reducible reflection 139, 152, 185, 186, 191-195,

401-406, 411, 418

reflection group 179, 185, 189, 191~195, 401~405

relatively minimal see elliptic surface, relatively minimal

removable singularity theorem see Uhlenbeck's removable singularity theorem

representable functor 304, 305, 334, 336

restricted Donaldson polynomial invariant see Donaldson polynomial invariant, restricted

ruled surface see surface, ruled ruling 20, 28, 59, 443

Salvetti, M. 410 Schwarzenberger, R.L.E. 488 secondary Hopf surface see surface,

Hopf Seifert fiber 7 Seiler's theorem 7, 16, 66~75, 82, 165 Seiler, W. 56, 57, 64 self-dual 2-form 12, 242, 255, 256, 319 self-dual curvature 247, 254, 255, 327 semistable bundle 323, 356, 359~361,

365, 366, 373 semistable point 359, 360, 362, 363 semistable torsion free sheaf 339 semi universal see locally semiuniversal Seshadri, C. 357 Shafarevich, I. 2, 14, 16, 19, 75,

84~86, 106 Shioda, T. 87 simple (O,I)-connection or vector

bundle 283, 294, 296, 300, 302~305, 314, 323, 325, 327, 336

simultaneous resolution of double points 33, 59, 94, 115, 164

singular support 244~246, 248, 249, 251, 254, 257, 263, 266, 347

slice 231, 232, 294~297, 306, 308, 326~331

slope 321 Smale, S. 1, 3 smooth divisor 268 Sobolev completion 227, 230, 293,

294, 365, 386 Sobolev embedding theorem 244 Sobolev multiplication theorem 241,

293 spherical orbifold see orbifold, spherical spin structure on a manifold 375~379

Spin(V) 375 spinor norm, real 191, 393, 397, 408,

411, 412, 414, 426

Index 519

spinors 235, 268, 375 stability of exceptional curves

see exceptional curves, stability of stable bundle 6, 9, 10, 279, 300, 303,

304, 320-323, 328~330, 334~338, 341, 342, 344, 366, 370

stable bundles over curves 356~366

stable Donaldson polynomial see Donaldson polynomial, stable

stable elements 229, 271~273, 276, 277

stable point 360, 361 stable range 9, 346, 352, 368, 373,

428 stable submanifold 111, 129 stratification, local 247~260

strongly minimal 4-manifold 419, 420, 499

strongly minimal model 419 sub-line bundle 321, 367, 445, 450,

453 surface, ruled 209 surface

Barlow 31, 498 Dolgachev 31 elliptic see elliptic surface Enriques 22, 23, 31 ~33, 50, 51, 131,

224, 225, 495 general type 4, 7, 23, 24, 28, 29,

31, 33, 155, 221, 409, 410, 494, 498, 499

geometrically ruled 20, 28, 29 Hopf 5,7, 17,20,51, 111, 132~137,

157, 199, 209, 221, 222, 224, 492, 494,495 primary 133~ 135, 137 secondary 133~ 135, 209

hyperelliptic 22, 23, 31, 32, 50, 51, 130, 131, 214, 215

Inoue 5, 20, 157 K3 3, 4, 22, 23, 31~33, 50, 51, 131,

132, 194, 224, 225, 408, 416, 442, 490, 491, 494~497

Kodaira 22, 23, 32, 33, 50, 130, 131, 214~216

rational 28, 31, 221, 225 rational ruled 18, 443, 444 ruled 20, 25, 27~31, 51, 59, 60, 131,

132, 199, 217, 222, 224, 420, 494 of Type VII 5, 14, 156, 157, 221

Tate-Shafarevich group, analytic 16, 75~82

Taubes' gluing theorem 228, 247, 253, 254,259

520 Index

Taubes, C. 240, 255, 259, 338 theta divisor 444, 486, 489 theta function 345 theta-characteristic 341-346, 352,

354, 461, 477, 480, 481 theta-characteristic, nontrivial 444,

469, 477, 484, 485 thickened completion see completion of

the moduli space, thickened Thurston, W. 1 triangle group 149-154,217 trivial connection see connection,

trivial Type VII see surface, Type VII Tyurin, A. 498 Tyurina, G. 32

Ue's theorem 5, 138, 159 Uhlenbeck compactification 244, 339,

346 Uhlenbeck limit 256, 275, 348,

436-438 Uhlenbeck's removable singularity

theorem 242 Uhlenbeck's weak compactness theorem

228, 239, 240, 246, 247, 370, 434 Uhlenbeck-Yau theorem 324 universal bundle 302, 303, 308,

334-337, 342-344, 364, 365, 459-461, 481, 482, 487, 490

universal elliptic curve 37-42, 124

universal SO(3)-bundle 227, 232, 233, 246, 331, 379

unobstructed see deformation, unobstructed

unstable point 359-361 upper half plane 36

Van de Ven, A. 5, 498 vanishing arcs 166-171, 183, 184 vanishing cycle 55,167-170,182, 183 vanishing disk 167, 183, 184, 188,

189, 190

Wall, C.T.C. 192, 194 weak compactness see Uhlenbeck's

weak compactness theorem weaker version of the ASD equations

228, 247, 255 Wehler, J. 133 Weierstrass model 16, 56-60, 62, 163,

164 Wood, J. 410

Yang-Mills equations 3 Yang-Mills functional 236 Yau, S.T. 31

Zariski tangent space 237, 324, 355 Zariski's main theorem 461 Zariski's theorem on minimal models

21

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Foige A Series of Modern Surveys in Mathematics

Ed.-in-chief: R. Remmert. Eds.: E. Bombieri, S. Feferman, M. Gromov, H. W. Lenstra, P-L.Lions, W.Schmid, J-P.Serre, J. Tits

Volume 1: A. Frohlich

Galois Module Structure of Algebraic Integers 1983. ISBN 3-540-11920-5

Volume 2: W. Fulton

Intersection Theory 1984. ISBN 3-540-12176-5

Volume 3:J. C.Jantzen

Einhullende Algebren halb­einfacher Lie-Algebren 1983. ISBN 3-540-12178-1

Volume 4: W. Barth, C. Peters, A. vandeVen

Compact Complex Surfaces 1984. ISBN 3-540-12172-2

Volume 5: K. Strebel

Quadratic Differentials 1984. ISBN 3-540-13035-7

Volume 6: M.J. Beeson

Foundations of Constructive Mathematics Metamathematical Studies

1985. ISBN 3-540-12173-0

Volume 8: R. Mane

Ergodic Theory and Differentiable Dynamics Translated from the Portuguese by Silvio Levy 1987. ISBN 3-540-15278-4

Volume 9: M. Gromov

Partial Differential Relations 1986. ISBN 3-540-12177-3

Volume 10: A. L. Besse

Einstein Manifolds 1986. ISBN 3-540-15279-2

Volume II: M. D. Fried, M.Jarden

Field Arithmetic 1986. ISBN 3-540-16640-8

Volume 12:J. Bochnak, M. Coste, M.-F.Roy

Geometrie algebrique reelle 1987. ISBN 3-540-16951-2

Springer B3 .IO.127

Volume 13: E. Freitag, R. Kiehl

Etale Cohomology and the Weil Conjecture With an Historical Introduction by]. A. Dieudonne

1987. ISBN 3-540-12175-7

Volume 14: M. R. Goresky, R. D. MacPherson

Stratified Morse Theory 1988. ISBN 3-540-17300-5

Volume 15: T.Oda

Convex Bodies and Algebraic Geometry An Introduction to the Theory of Toric Varieties

1987. ISBN 3-540-17600-4

Volume 16: G. van der Geer

Hilbert Modular Surfaces 1988. ISBN 3-540-17601-2

Volume 17: G. A. Margulis

Discrete Subgroups of Semisimple Lie Groups 1990. ISBN 3-540-12179-X

Volume 18: A.E.Brouwer, A.M. Cohen, A.Neumaier

Distance-Regular Graphs 1989. ISBN 3-540-50619-5

Volume 19: I. Ekeland

Convexity Methods in Hamiltonian Mechanics 1990. ISBN 3-540-50613-6

Volume 20: A.I. Kostrikin

Around Burnside 1990. ISBN 3-540-50602-0

Volume 21: S. Bosch, W. Liitkebohmert, M.Raynaud

Neron Models 1990. ISBN 3-540-50587-3

Volume 22: G. Faltings, C.-L. Chai

Degeneration of Abelian Varieties 1990. ISBN 3-540-52015-5

Volume 23: M. Ledoux, M. Talagrand

Probability in Banach Spaces Isoperimetry and Processes

1991. ISBN 3-540-52013-9

Volume 24: V. F. Lazutkin

KAM Theory and Semi­classical Approximations to Eigenfunctions Appendix by A. I. Shnirelman 1993. ISBN 3-540-53389-3

Volume 25: W. de Melo, S. van Strien

One-Dimensional Dynamics 1993. ISBN 3-540-56412-8

Volume 26: S. Rickman

Quasiregular Mappings 1993. ISBN 3-540-56648-1

Springer 8.1.ID.1l7