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Abstract

In this dissertation, the main objects of study are sheaves with a 'twisted endomorphism' on surfaces with an automorphism. The results are divided into three main chapters: In Chapter 2, we work generally and define these objects on varieties with an automorphism. The moduli stacks of these objects are shown to be algebraic. When the automorphism is of finite order, we show that the category of such objects is equivalent to a category of modules over a certain sheaf of noncommutative algebras. After fixing a stability condition, we study Hilbert schemes of points of this sheaf of noncommutative algebras. In Chapter 3, we specialize to the case of surfaces. The topological Euler characteristics of the Hilbert schemes in Chapter 2 are packaged in a generating series, which we compute. The answer generalizes the situation when the automorphism is the identity. Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) type invariants arising from these objects. We define DT-type invariants when the structure sheaf of the surface has no higher cohomology and the quotient map is a cyclic covering of smooth surfaces. This gives new examples of invariants for automorphisms of some del Pezzo surfaces, which also may be thought of as invariants of noncommutative Fano threefolds.