This course is devoted to studying topological properties of complex algebraic varieties using arithmetic methods. At first we will review the (proven) Weil conjectures which relate geometry over finite fields and the complex numbers. Subsequently we will turn to p-adic integration and prove Batyrev's theorem: Birational Calabi-Yau varieties have equal Betti numbers. If time permits we will briefly encounter crepant resultions and the McKay correspondence and conclude the course by discussing purely complex geometric alternatives to those arithmetic techniques: mixed Hodge structures and motivic integration.
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