Chapter 1 Introduction ......................................... 1
1.1 Methods .................................................... 2
Chapter 2 Background ........................................... 5
2.1 Connections ................................................ 5
2.2 Sheaves and cohomology ..................................... 7
2.3 Toric manifolds ............................................ 9
2.4 Geometric quantization and polarizations .................. 10
2.5 Examples .................................................. 13
2.6 Aside: Rigidity of Bohr-Sommerfeld leaves ................. 14
Chapter 3 The cylinder ........................................ 15
3.1 Flat sections and Bohr-Sommerfeld leaves .................. 15
3.2 Sheaf cohomology .......................................... 16
3.3 Brick wall covers ......................................... 19
3.4 Mayer-Vietoris ............................................ 24
3.5 Refinements and covers: Scaling the brick wall ............ 26
Chapter 4 The complex plane ................................... 29
4.1 The sheaf of sections flat along the leaves ............... 29
4.2 Cohomology ................................................ 30
4.3 Mayer-Vietoris ............................................ 34
Chapter 5 Example: S2 ......................................... 35
Chapter 6 The multidimensional case ........................... 37
6.1 The model space ........................................... 37
6.2 The flat sections ......................................... 37
6.3 Multidimensional Mayer-Vietoris ........................... 38
Chapter 7 A better way to calculate cohomology ................ 41
7.1 Theory .................................................... 41
7.2 The case of one dimension ................................. 45
7.3 The structure of the coming calculation ................... 45
7.4 The case of several dimensions: Non-singular .............. 46
7.5 The partially singular case ............................... 49
Chapter 8 Piecing and glueing ................................. 51
8.1 Necessary sheaf theory .................................... 51
8.2 The induced map on cohomology ............................. 52
8.3 Patching together ......................................... 54
Chapter 9 Real and Kahler polarizations compared .............. 57
Bibliography ................................................... 59
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