Summary ......................................................... 1
Introduction .................................................... 4
1 Deformation theoretic approach .............................. 11
1.1 Hodge structures of КЗ type ............................ 11
1.1.1 Hodge structures ................................ 11
1.1.2 Hodge structures of КЗ type ..................... 14
1.1.3 Endomorphisms of Г .............................. 16
1.1.4 Mukai's result and КЗ surfaces with CM .......... 16
1.1.5 Splitting of T over extension fields ............ 17
1.1.6 Galois action on TF ............................. 18
1.1.7 Weil restriction ................................ 19
1.1.8 The special Mumford Tate group of T ............. 20
1.2 The variational approach ............................... 22
1.2.1 The Hodge locus of an endomorphism .............. 22
1.2.2 Proof and discussion of Theorem 1 ............... 24
1.2.3 Twistor lines ................................... 28
2 The Kuga-Satake correspondence .............................. 31
2.1 Kuga Satake varieties and real multiplication .......... 31
2.1.1 Clifford algebras ............................... 31
2.1.2 Spin group and spin representation .............. 32
2.1.3 Graded tensor product ........................... 32
2.1.4 Kuga Satake varieties ........................... 33
2.1.5 Constriction of algebras ........................ 33
2.1.6 The decomposition theorem ....................... 36
2.1.7 Galois action on C(g) .......................... 37
2.1.8 Proof of the decomposition theorem .............. 39
2.1.9 Central simple algebras ......................... 45
2.1.10 An example ...................................... 46
2.2 Double covers of  2 branched along six lines ........... 49
2.2.1 The transcendental lattice ...................... 49
2.2.2 Moduli .......................................... 50
2.2.3 Endoniorphisms of the transcendental lattice .... 51
2.2.4 Abelian varieties of Weil type .................. 52
2.2.5 Abelian varieties with quaternion
multiplication .................................. 53
2.2.6 The Kuga-Satake variety ......................... 53
2.2.7 Proof of Theorem 2 .............................. 54
3 Hilbert schemes of points on КЗ surfaces .................... 56
3.1 The cohoniology of the Hilbert square .................. 57
3.1.1 The cohoniology ring ............................ 57
3.1.2 HC for S×S ⇔ HC for Hilb2(S) .................... 60
3.2 Tautological bundles on the Hilbert square ............. 62
3.2.1 The fundamental short exact sequence ............ 62
3.2.2 The Chern character of  |2| ..................... 63
3.2.3 The stability of |2| ........................... 66
3.3 The Fano variety of lines on a cubic fourfold .......... 72
3.3.1 The result of Beauville and Donagi .............. 72
3.3.2 Chern classes of F .............................. 73
3.3.3 The image of the correspondence [Z]* ............ 75
3.3.4 The Fano surface of lines on a cubic
threefold ....................................... 76
3.4 Discussion ............................................. 77
4 Two complementary results ................................... 79
4.1 КЗ surfaces with CM are defined over number fields ..... 79
4.2 Andre motives .......................................... 82
4.2.1 Tensor categories and Tannakian categories ...... 82
4.2.2 Andre motives ................................... 83
4.2.3 Markman's results ............................... 87
4.2.4 The motive of X ................................. 90
4.2.1 Bibliography .................................... 93
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