A. Introduction ................................................. 7
A.l. State of the art ........................................ 7
A.1.1. Contact manifolds and quaternion-Kähler
manifolds ........................................ 7
A.1.2. Lines on contact Fano manifold ................... 8
A.1.3. Legendrian subvarieties of projective space ..... 10
A.2. Topics of the dissertation ............................. 12
A.3. Open problems .......................................... 16
B. Notation and elementary properties .......................... 17
B.l. Vector spaces and projectivisation ..................... 17
B.2. Bilinear forms and their matrices ...................... 17
B.3. Complex and algebraic manifolds ........................ 17
B.4. Vector bundles, sheaves and sections ................... 18
B.5. Derivatives ............................................ 18
B.6. Homogeneous differential forms and vector fields ....... 19
B.7. Submersion onto image .................................. 19
B.8. Tangent cone ........................................... 20
B.9. Secant, tangent and dual varieties ..................... 20
С Vector fields, forms and automorphisms ...................... 21
C.l. Vector fields, Lie bracket and distributions ........... 21
C.2. Automorphisms .......................................... 23
C.3. Line bundles and  *-bundles ............................ 24
C.4. Distributions and automorphisms preserving them ........ 25
C.5. Lifting and descending twisted forms ................... 26
D. Elementary symplectic geometry .............................. 29
D.l. Linear symplectic geometry ............................. 29
D.l.l. Symplectic vector space ......................... 29
D.l.2. Subspaces in a symplectic vector space .......... 30
D.1.3. Symplectic reduction of a vector space .......... 30
D.1.4. Symplectic automorphisms and weks-symplectic
matrices ........................................ 31
D.1.5. Standard symplectic structure on W ⊕ W* ......... 32
D.2. Symplectic manifolds and their subvarieties ............ 32
D.2.1. Subvarieties of a symplectic manifold ........... 32
D.2.2. Examples ........................................ 33
D.3. The Poisson bracket .................................... 35
D.3.1. Properties of Poisson bracket ................... 36
D.3.2. Homogeneous symplectic form ..................... 37
D.3.3. Example: Veronese map of degree 2 ............... 37
E. Contact geometry ............................................ 38
E.l. Projective space as a contact manifold ................. 39
E.2. Legendrian subvarieties of projective space ............ 40
E.2.1. Decomposable and degenerate Legendrian
subvarieties .................................... 40
E.3. Contact manifolds ...................................... 41
E.3.1. Symplectisation ................................. 41
E.3.2. Contact automorphisms ........................... 42
E.4. Legendrian subvarieties in contact manifolds ........... 44
F. Projective automorphisms of a Legendrian variety ............ 47
F.l. Discussion of assumptions .............................. 48
F.2. Preservation of contact structure ...................... 48
F.3. Some comments .......................................... 51
G. Toric Legendrian subvarieties in projective space ........... 52
G.l. Classification of toric Legendrian varieties ........... 52
G.2. Smooth toric Legendrian surfaces ....................... 54
G.3. Higher dimensional toric Legendrian varieties .......... 56
H. Examples of quasihomogeneous Legendrian varieties ........... 59
H.l. Notation and definitions ............................... 59
H.2. Main results ........................................... 61
H.2.1. Generalisation: Representation theory and
further examples ................................ 61
H.3. G-action and its orbits ................................ 63
H.3.1. Invariant subsets ............................... 65
H.3.2. Action of  ..................................... 67
H.4. Legendrian varieties in Y .............................. 67
H.4.1. Classification .................................. 67
H.4.2. Degenerate matrices ............................. 68
H.4.3. Invertible matrices ............................. 69
I. Hyperplane sections of Legendrian subvarieties .............. 75
I.1. Hyperplane section ..................................... 76
I.1.1. Construction .................................... 76
I.1.2. Proof of smoothness ............................. 76
I.2. Linear sections of decomposable Legendrian varieties ... 78
I.3. Extending Legendrian varieties ......................... 80
I.4. Smooth varieties with smooth dual ...................... 83
References ..................................................... 84
|
