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Theory and Applications of Holomorphic Functions on Algebraic Varieties over Arbitrary Ground Fields
eBook ISBN:  9780821899946 
Product Code:  MEMO/1/5.E 
List Price:  $21.00 
MAA Member Price:  $18.90 
AMS Member Price:  $16.80 
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Theory and Applications of Holomorphic Functions on Algebraic Varieties over Arbitrary Ground Fields
eBook ISBN:  9780821899946 
Product Code:  MEMO/1/5.E 
List Price:  $21.00 
MAA Member Price:  $18.90 
AMS Member Price:  $16.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 1; 1951; 90 ppMSC: Primary 14

Table of Contents

Chapters

Introduction

Part I. General theory of holomorphic functions

1. Strongly holomorphic functions

2. Strongly holomorphic functions on affine models

3. The general concept of a holomorphic function

4. Rational holomorphic functions. Some unsolved problems

5. Digression on algebraic points

6. Holomorphic functions, analytical irreducibility and a connectedness criterion

7. Analytical irreducibility and normalization

8. Some lemmas on $\mathfrak {m}$adic rings

9. Holomorphic functions on affine models

Part II. Invariance of rings of holomorphic functions under rational transformations

10. Holomorphic functions and semiregular birational transformations

11. Absolute birational invariance of rings of holomorphic functions

12. Reduction of the proof of the fundamental theorem to a special case

13. The birational transformation $V \dashrightarrow V\circ t$

14. Proof of the fundamental theorem in the case of the transformation $V \dashrightarrow V\circ t$

15. Last step of the proof of the fundamental theorem: transition to the derived normal model $\overline {V\circ t}$

16. Extension of the fundamental theorem to rational transformations

17. Reduction of the proof to a special case

18. The rational transformationt $V \dashrightarrow V\circ t$

19. The transformation $V\circ t \dashrightarrow \overline {V\circ t}$

20. A connectedness theorem for algebraic correspondences

21. Algebraic systems of $r$cycles

22. Proof of the principle of degeneration


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Chapters

Introduction

Part I. General theory of holomorphic functions

1. Strongly holomorphic functions

2. Strongly holomorphic functions on affine models

3. The general concept of a holomorphic function

4. Rational holomorphic functions. Some unsolved problems

5. Digression on algebraic points

6. Holomorphic functions, analytical irreducibility and a connectedness criterion

7. Analytical irreducibility and normalization

8. Some lemmas on $\mathfrak {m}$adic rings

9. Holomorphic functions on affine models

Part II. Invariance of rings of holomorphic functions under rational transformations

10. Holomorphic functions and semiregular birational transformations

11. Absolute birational invariance of rings of holomorphic functions

12. Reduction of the proof of the fundamental theorem to a special case

13. The birational transformation $V \dashrightarrow V\circ t$

14. Proof of the fundamental theorem in the case of the transformation $V \dashrightarrow V\circ t$

15. Last step of the proof of the fundamental theorem: transition to the derived normal model $\overline {V\circ t}$

16. Extension of the fundamental theorem to rational transformations

17. Reduction of the proof to a special case

18. The rational transformationt $V \dashrightarrow V\circ t$

19. The transformation $V\circ t \dashrightarrow \overline {V\circ t}$

20. A connectedness theorem for algebraic correspondences

21. Algebraic systems of $r$cycles

22. Proof of the principle of degeneration
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