Cyclic Coverings, Calabi-yau Manifolds And Complex Multiplication (lecture Notes In Mathematics)
by Christian Rohde /
2009 / English / PDF
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Calabi-Yau manifolds have been an object of extensive research
during the last two decades. One of the reasons is the importance
of Calabi-Yau 3-manifolds in modern physics - notably string
theory. An interesting class of Calabi-Yau manifolds is given by
those with complex multiplication (CM). Calabi-Yau manifolds with
CM are also of interest in theoretical physics, e. g. in connection
with mirror symmetry and black hole attractors. It is the main aim
of this book to construct families of Calabi-Yau 3-manifolds with
dense sets of ?bers with complex multiplication. Most - amples in
this book are constructed using families of curves with dense sets
of ?bers with CM. The contents of this book can roughly be divided
into two parts. The ?rst six chapters deal with families of curves
with dense sets of CM ?bers and introduce the necessary theoretical
background. This includes among other things several aspects of
Hodge theory and Shimura varieties. Using the ?rst part, families
of Calabi-Yau 3-manifolds with dense sets of ?bers withCM are
constructed in the remaining ?ve chapters. In the appendix one ?nds
examples of Calabi-Yau 3-manifolds with complex mul- plication
which are not necessarily ?bers of a family with a dense set ofCM
?bers. The author hopes to have succeeded in writing a readable
book that can also be used by non-specialists.
Calabi-Yau manifolds have been an object of extensive research
during the last two decades. One of the reasons is the importance
of Calabi-Yau 3-manifolds in modern physics - notably string
theory. An interesting class of Calabi-Yau manifolds is given by
those with complex multiplication (CM). Calabi-Yau manifolds with
CM are also of interest in theoretical physics, e. g. in connection
with mirror symmetry and black hole attractors. It is the main aim
of this book to construct families of Calabi-Yau 3-manifolds with
dense sets of ?bers with complex multiplication. Most - amples in
this book are constructed using families of curves with dense sets
of ?bers with CM. The contents of this book can roughly be divided
into two parts. The ?rst six chapters deal with families of curves
with dense sets of CM ?bers and introduce the necessary theoretical
background. This includes among other things several aspects of
Hodge theory and Shimura varieties. Using the ?rst part, families
of Calabi-Yau 3-manifolds with dense sets of ?bers withCM are
constructed in the remaining ?ve chapters. In the appendix one ?nds
examples of Calabi-Yau 3-manifolds with complex mul- plication
which are not necessarily ?bers of a family with a dense set ofCM
?bers. The author hopes to have succeeded in writing a readable
book that can also be used by non-specialists.
