introduction

historical note

the contents of the book

standard notation

i.preliminaries

topology and algebra

1.notation and basic facts

2.some properties of bilinear forms

3.vector bundles, characteristic classes and the index
theorem

complex manifolds

4.basic concepts and facts

5.holomorphic vector bundles, serre duality and riemann-roch

6.line bundles and divisors

7.algebraic dimension and kodaira dimension

general analytic geometry

8.complex spaces

9.the a-process

10.deformations of complex manifolds

differential geometry of complex manifolds

11.de rham cohomology

12.dolbeault cohomology

13.kaihler manifolds

14.weight-1 hodge structures

15.yau's results on kaihler-einstein metrics

coverings

16.ramification

17.cyclic coverin

18.covering tricks

projective-algebraic varieties

19.gaga theorems and projectivity criteria

20.theorems of bertini and lefschetz

ii.curves on surfaces

embedded curves

1.some standard exact sequences

2.the picard-group of an embedded curve

3.piemann-roch for an embedded curve

4.the residue theorem

5.the trace map

6.serre duality on an embedded curve

7.the a-process

8.simple singularities of curves

intersection theory

9.intersection multiplicities

10.intersection numbers

11.the arithmetical genus of an embedded curve

12.1-connected divisors

iii.mappings of surfaces

bimeromorphic geometry

1.bimeromorphic maps

2.exceptional curves

3.rational singularities

4.exceptional curves of the first kind

5.hirzebruch-jung singularities

6.resolution of surface singularities

7.singularities of double coverings, simple singularities of
surfaces

fibrations of surfaces

8.generalities on fibratious

9.the n-th root fibration

10.stable fibrations

11.direct image sheaves

12.relative duality

the period map of stable fibrations

13.period matrices of stable curves

14.topological monodromy of stable fibrations

15.monodromy of the period matrix

16.extending the period map

17.the degree of f.ωx/s

18.iitaka's conjecture c2,1

iv.some general properties of surfaces

1.meromorphic maps, associated to line bundles

2.hodge theory on surfaces

3.existence of kahler metrics

4.deformations of surfaces

5.some inequalities for hodge numbers

6.projectivity of surfaces

7.the nef cone

8.surfaces of algebraic dimension zero

9.almost-complex surfaces without any complex structure

10.bogomolov's theorem

11.reider's method

12.vanishing theorems on surfaces

v.examples

some classical examples

1.the projective plane p2

2.complete intersections

3.tori of dimension 2

fibre bundles

4.ruled surfaces

5.elliptic fibre bundles

6.higher genus fibre bundles

elliptic fibrations

7.kodaira's table of singular fibres

8.stable fibrations

9.the jacobian fibration

10.stable reduction

11.classification

12.invariants

13.logarithmic transformations

kodaira fibrations

14.kodaira fibrations

finite quotients

15.the godeaux surface

16.kummer surfaces

17.quotients of products of curves

infinite quotients

18.hopf surfaces

19.inoue surfaces

20.quotients of bounded domains in c:

21.hilbert modular surfaces

coverings

22.invariants of double coverings

23.an enriques surface

24.kummer coverings

vi.the enriques kodaira classification

1.statement of the main result

2.characterising minimal surfaces whose canonical bundle is
nef

3.the rationality theorem and castelnuovo's criterion

4 the casea(x)= 2

5.the casea(x)= 1

6.thecase a(x)=0

7.the final step

8.deformations

vii.surfaces of general type

preliminaries

1.introduction

2.some general theorems

two inequalities

3.noether's inequality

4.the inequality c＜3c2

pluricanonical maps

5.the main results

6.proof of the main results

7.the exceptional cases and the 1-canonical map

surfaces with given chern numbers

8.the geography of chern numbers

9.surfaces on the noether lines

10.surfaces with q = pg = 0

viii.k3-surfaces and enrlques surfaces

introduction

1.notation

2.the results

k 3-surfaces

3.topological and analytical invariats

4.digresmon on affine geometry over f2

5.the neron-severl lattice of kummer surfaces

6.the tore]ii theorem for kummer surfaces

7.the local torelli theorem for k3-surfaces

8.a density theorem

9.behaviour of the kehler cone under deformations

10.degenerations of isomorphisms between k3-surfaces

11.the toreui theorems for k3-surfaces

12.construction of moduli spaces

13.digression on quaternionic structures

14.surjectivity of the period map

enriques surfaces

15.topological and analytic invariants

16.divisors on an enriques surface y

17.elliptic pencils

18.double coverings of quadrics

19.the period map

20.the period domain for enriques surfaces

21.global properties of the period map special topics

22.projective k3-surfaces and mirror symmetry

23.special curves on k3-surfaces

24.an application to hyperbolic geometry

ix.topological and differentiable structure of surfaces

topology of simply connected compact complex surfaces

1.freedman's results

2.representability of unimodular forms

donaldson invaxiants

3.introduction

4.the donaldson invariant, a bird's eye view

5.infinitely many homeomorphic surfaces which are not diffeomorphic
..

6.further results obtained by the donaldson method

seiberg-witten invariants

7.introduction

8.properties of the invariants

9.surfaces diffeomorphic to a rational surface

bibliography

notation

index