Following D. Huybrechts, Complex Geometry, Springer 2004

1. SEMINAR ON COMPLEX GEOMETRY
PROF. DANIEL HUYBRECHTS, HEINRICH HARTMANN
In this seminar we want to go through the basic theory of complex man-ifolds
as presented in the book [2].
In order to stimulate active participation, we will organize the seminar as a
reading class. This means that everyone should read the announced material
each time. We will pick out some particularly interesting topics which are
presented by a speaker. This means the remaining material should be read
with more care. At the meeting we will reserve half an hour for a discussion,
where we can answer questions about the reading material. Sometimes we
may also provide some exercises, which we will then solve in a common
effort. In the remaining hour we will have the talk.
In the program below, the section ”Scope” refers to the material that we
want to learn/read this week. The section ”Talk” lists the topics which are
to be presented by the speaker. All references will refer to [2].
I. Local Theory
1. Holomorphic Functions of Several Variables.
Scope: p.1-9. Comparison of function theory in one variable to function
theory of several variables.
Talk: Recall the definition of a holomorphic function in several variables.
Use the Cauchy integral formula to prove that they admit a power series ex-pansion.
Prove Hartogs theorem and the Weierstrass preparation theorem.
2. Local Theory of Holomorphic Functions.
Scope: p.10-17. Inverse and implicit function theorems. Basic properties
of the local ring R = OCn,0.
Talk: Define the sheaf of holomorphic functions. Prove that R is a unique
factorization domain (UFD), and that it is noetherian. For the latter we will
use the Weierstrass division theorem (do not prove it). Furthermore give a
proof of corollary. 1.1.19: If f vanishes on Z(g), g irreducible, then g|f.
II. Complex manifolds and holomorphic vector bundles
3. Complex Manifolds.
Scope: p.52-63. Definition of complex manifolds and holomorphic maps
between them. Siegel’s theorem. Examples of complex manifolds.
Talk: Prove Siegel’s theorem (2.1.9). Show how to obtain charts for an
affine hypersurface f−1(0) ⊂ Cn from the implicit function theorem. Explain
the ideal sheaf associated to a submanifold/subvariety, and the associated
structure sequence.
Date: January 8, 2010.
1

2. 2 PROF. DANIEL HUYBRECHTS, HEINRICH HARTMANN
Intermezzo: Sheaf Cohomology (skipped).
This material should was well known to the participants of the algebraic
geometry course last semester so it was possible to skip this topic.
Scope: Appendix B. The focus lies on basic definitions (sheaves, stalks,
sheafification, exact sequences) and ˇ Cech cohomology.
Talk: Recall definition of stalks and exact sequences of sheaves. Prove
B.0.28: A complex is exact iff it is exact on the stalks. Define ˇ Cech Co-homology.
Introduce the long exact sequence of ˇ Cech cohomology groups
associated to a short exact sequence of sheaves. Give an explicit description
of the boundary map.
4. Holomorphic Vector Bundles.
Scope: p.66-72. Definition of holomorphic vector bundles. Standard con-structions
(direct sums, tensor products, etc.). Cocycle descriptions. Picard
group. The first Chern class. Holomorphic tangent bundle. Adjunction
formula. Vector bundles and locally free sheaves.
Talk: Prove Corollary 2.2.10: Pic(X) ∼=
H1(X,O
X). Explain how to get
the first Chern class form the exponential sequence. Prove the Adjunction
formula (2.2.17).
5. Divisors and Line Bundles.
Scope: p.77-84. Weil-divisors. Cartier-divisors. Associated line bundles.
Functorial properties. Linear Equivalence. Meromorphic sections. Also we
need some more background from chapter 1 (p.18-20) on (singular) analytic
hypersurfaces, in order to define Weil-divisors. For the first reading one can
restrict to smooth case, where this is not needed.
Talk: Prove the equivalence of Weil- and Cartier-divisors (Proposition
2.3.9). Prove Proposition 2.3.18 and Exercise 2.3.2 concerning the vanishing
of sections of line bundles.
6. Projective Space.
Scope: p.85-97. Maps to projective space associated to a line bundle.
Abel-Jacobi map. Tautological bundle on the projective space. Canonical
bundle. Euler sequence.
Talk: Exercise 2.3.7. Proposition 2.4.1. on the global sections of OPn(k).
Proposition 2.4.4. (Euler Sequence) or Exercise 2.4.10 generalization to
Gassmanians.
7. Differential Forms on Complex Manifolds.
Scope: Linear algebra of differential forms on p.25-28: Almost complex
structures and the decomposition of the exterior algebra VC (Proposition
1.2.8). Sections 1.3 and 2.6: Real- and holomorphic tangent bundle. @, ¯@-
operators. ¯@-Poincare lemmas. Dolbeaut cohomology groups.
Talk: Prove the ¯@-Poincare lemmas 1.3.7 and 1.3.8. Prove proposition
X to Ap,0
X . Define the Dolbeaut complex and cohomology
2.6.11 relating
p
groups. Deduce corollary 2.6.25. from the above statements.

3. SEMINAR ON COMPLEX GEOMETRY 3
III. K¨ahler manifolds
8. K¨ahler Manifolds.
Scope: Basic hermitian linear algebra p.28-29: Compatible scalar prod-ucts
and the fundamental form. Local theory p.48-49: A K¨ahler metric
osculates to order two. Moreover definition of a K¨ahler manifold and the
computations for the projective space (p.118-119).
Talk: We want to see the computation of the Fubini–Study metric on Pn
in full detail (p.117-119). Furthermore show that RPn !FS = 1 at least for
P1 (Exercise 3.1.4.). Prove that any projective manifold is K¨ahler (3.1.10,
3.1.11).
9. Hermitian Linear Algebra.
Scope: Section 1.2. p.30-40: Properties of vector spaces with almost
complex structure and compatible scalar product. Lefschetz operator and
Hodge-∗ operator on VC. Weil’s theorem 1.2.31. Hodge–Riemann bilinear
relations (linear algebra version).
Talk: Recall the definition of the Lefschetz operator and its dual. Prove
that they define a sl2 representation on VC (Proposition 1.2.26) and the
Lefschetz decomposition theorem (1.2.30).
10. K¨ahler Identities.
Scope: Section 3.1. Laplace operators, K¨ahler identities.
Talk: Give an overview over the definitions of the operators occuring in
the statement of the K¨ahler identities (3.1.12) and prove them.
11. Hodge Decomposition (8.1.2010).
Scope: Section 3.2. Definition of d, @, ¯@-harmonic forms. Serre duality and
Poincare duality for harmonic forms. @ ¯@-Lemma. Hodge decomposition in
cohomology.
Talk: Prove Proposition 3.2.6 about d, @, ¯@-harmonic forms. Prove corol-lary
3.2.12: The Hodge decomposition does not depend on the metric. Show
how remark 3.2.7. implies the corresponding symmetries of the cohomology
groups (see also the diagram on page 138).
12. Lefschetz Theorems (15.1.2010).
Scope: Section 3.3. p.132-137. Lefschetz theorem on (1,1) classes. Neron–
Severi group. Jacobian and Albanese Tori. Hard Lefschetz theorem.
Talk: Prove the Lefschetz theorem on (1,1) classes (Lemma 3.3.1, Propo-sition
3.2.2). Show how the Hard Lefschetz theorem (3.3.13) can be derived
form the decomposition of the exterior algebra introduced earlier.
13. Hodge Theorems (22.1.2010).
Scope: Section 3.3. p.138-142. Hodge–Riemann bilinear relations. Hodge
index theorem. Signature formula.
Talk: Prove the Hodge index theorem (3.3.16) and the signature formula
(3.3.17).

4. 4 PROF. DANIEL HUYBRECHTS, HEINRICH HARTMANN
IV. Curves
In this last part of the seminar we want to apply the theory developed so
far to the one-dimensional case. This allows us to see more explicitly how
those techniques work. Moreover we will deduce the Riemann–Roch and the
Serre duality theorem for vector bundles on curves. These have analogues
for higher dimensional manifolds, which are more complicated to prove.
We will use the book [1], Chapter VII, as reference. You will notice, that
we did not cover much of the chapters I-VI of this book. Nevertheless it
should be possible to read this chapter directly. The main theorems that
will be quoted are the finite-dimensionality of the cohomology of coherent
sheaves, and the vanishing of the cohomology on Stein spaces.
We have already met the finite-dimensionality in the case of the coherent
sheaves
i
X. The general case is very similar. Also the vanishing was shown
in talk 7 in the case of X = Cn.
14. Riemann–Roch Theorem (29.1.2010).
Scope: Paragraphs 1-3 of Chapter VII in [1]. The first paragraph recalls
the basic facts about divisors and locally free sheaves (line bundles) in the
case of curves. Please read it carefully, to get used to the (slightly different)
language used in this book.
Further contents: Existence of global meromorphic sections, Euler–Poincare
characteristic, vanishing of H2, Riemann–Roch.
Talk: Present paragraphs 2-3 culminating in the proof of the Riemann–
Roch theorem for divisors. If time permits you can also mention its gener-alization
to arbitrary vector bundles in the appendix to paragraph 4.
15. Serre Duality (5.2.2010).
Scope: Paragraphs 4-6 of Chapter VII in [1]. In the paragraphs 4 and 5 it
is proven that every vector bundle has a sub-bundle of the form O(D) and
that H1(X,M) = 0 for a curve X and its sheaf of meromorphic functions
M. Both statements should be well known in the algebraic context. We
will not have time to present them in the talk.
Talk: Present the proof of Serre duality
H0(X,
X(D))) = H1(X,O(−D))
in paragraph 6. We have already seen a version of this for the sheaves
i on
a general manifold (using Hodge ∗-operator and harmonic representatives).
This proof is more explicit/geometric and uses residues of differential forms.
It goes as follows:
We use a special “resolution” of the sheaf O(D) to write H1(X,O(D))
as quotient of two infinite–dimensional vector spaces of “sums of germs”
of local sections R/(R(D) +M(X)). Any meromorphic differential form !
determines a residue map !∗ : R7→ C, which is shown to induce a linear
form on H1(X,O(D)) = R/(R(D) +M(X)). The last step is to show that
the restriction to meromorphic differential forms ! ∈
X(−D) gives an
isomorphism H0(X,
X(−D)) → H1(X,O(D)).

5. SEMINAR ON COMPLEX GEOMETRY 5
References
[1] H. Grauert and R. Remmert, Theory of Stein spaces, Classics in Mathematics,
Springer 2004
[2] D. Huybrechts, Complex Geometry, Springer 2004