Complex Geometry Facts

Question 1

Let f:X -> Y be a nonconstant holomorphic map between Riemann surfaces. Is f an open map, closed map, or neither?

Answer 1

f is open

Question 2

Let f:X -> Y be an injective holomorphic map between Riemann surfaces. Then f is a biholomorphic mapping of X onto f(X) true or false?

Answer 2

True

Question 3

State the maximum principle for Riemann surfaces

Answer 3

Suppose X is a Riemann surfaces and f:X-> C is a non-constant holomorphic function. Then the absolute value of f does not attain its maximum.

Question 4

Let f:X -> Y be a nonconstant holomorphic map between Riemann surfaces. Suppose X is compact. What can we say about Y and f?

Answer 4

Then Y is compact and f is surjective.

Question 5

What can we say about holomorphic functions on compact Riemann surfaces?

Answer 5

They are constant.

Question 6

What can we say about meromorphic functions f on P^1?

Answer 6

They are rational functions, i.e., quotient of two polynomials.

Question 7

What is Liouville’s theorem?

Answer 7

Every bounded holomorphic function f:C -> C is constant.

Question 8

What can we say about doubly periodic holomorphic functions f:C->C?

Answer 8

They are constant.

Question 9

What can we say about nonconstant doubly periodic meromorphic functions f:C->P^1?

Answer 9

They attain every value c in P^1.

Question 10

True or false: Any connected proper open subset of C is biholomorphic to the open unit disk

Answer 10

False. We need simply connected assumption instead of connected.

Question 11

State Runge’s theorem.

Answer 11

Any holomorphic function on a compact set with connected complement can be uniformly approximated by polynomials.

Question 12

State Schwarz’s lemma.

Answer 12

If f:D ->D is a holomorphic map with f(0)=0, then |f(z)| \leq |z| and |f’(0)|\leq 1. Equality holds iff f is a rotation.

Question 13

What is the definition of the fundamental group of a topological space X?

Answer 13

The group of homotopy classes of loops based at some point in X.

Question 14

What is the fundamental group of S^1?

Answer 14

The group of integers (Z,+).

Question 15

What is the fundamental group of RP^2

Answer 15

Z/2Z.

Question 16

What is the singular homology of the n-sphere?

Answer 16

H_k(S^n) = Z, when k=0,n
0, otherwise.

Question 17

What is a stable point in the sense of GIT?

Answer 17

Points with closed orbits and finite stabilisers.

Question 18

State Riemann’s removable singularity theorem

Answer 18

Let U be an open subset of a Riemann surface and let a\in U. Suppose the function f \in O(U{a}) is bounded in some neighbourhood of a. Then f can be extended uniquely to a function F \in O(U).

Question 20

State the identity theorem

Answer 20

Suppose X and Y are Riemann surfaces and f_1, f_2: X \to Y are two holomorphic mappings which coincide on a set A\subset X having a limit point a\in X. Then f_1,f_2 are identical.

Question 21

Let f be a complex valued function on P^1. Then f is holomorphic at infinity if and only if what?

Answer 21

f(1/z) is holomorphic at z=0

Question 22

A closed subset of a compact space is what?

Answer 22

Compact

Question 23

A compact subset of a Hausdorff space is?

Answer 23

Closed

Question 24

A finite union of compact spaces is?

Answer 24

Compact

Question 25

Why are singleton sets {x} in Hausdorff topological spaces X closed?

Answer 25

The complement is closed because for every point y in the complement X\{x} there’s exists open neighbourhoods which seperate {x} and {y}.

Question 26

Suppose X and Y are locally compact spaces and p: X \to Y is a proper local homeomorphism. What is the conclusion?

Answer 26

Then p is a covering map.

Question 27

What is the principle of isolated zeros?

Answer 27

The zeros of an analytic function are isolated.