Abstract:

Moduli spaces are fundamental objects of interest across mathematics and physics. These spaces parametrize isomorphism classes of algebro-geometric objects, which come about frequently as solutions to classification problems. In algebraic geometry, of special interest are moduli spaces of coherent sheaves on smooth projective varieties. A fruitful approach to studying these moduli spaces is via Fourier-Mukai transforms on the bounded derived category D^b(CohX), built as the category of complexes of coherent sheaves on X, up to quasi-isomorphism. But these autoequivalences do not always take sheaves to sheaves. Enter Bridgeland's stability conditions - these conditions, whose origins began in string theory, allow us to apply autoequivalences freely and use our intuition from classical stability. Under suitable assumptions, for each stability condition and each numerical class, moduli spaces of stable objects in D^b(X) for an algebraic variety X, exist as proper algebraic spaces. This formalism includes many previously studied moduli spaces such as moduli spaces of Giesker or slope-stable sheaves. For most applications of Bridgeland stability, one needs projective coarse moduli spaces of Bridgeland semistable objects. One problematic aspect of Bridgeland stability is showing that the proper algebraic spaces above are in fact projective varieties. Many varieties come about as quotients of covering varieties by the action of some group G. We prove the following conjecture - Let X be a smooth projective variety, and G a finite cyclic group acting on X. Assume X carries projective Bridgeland semistable moduli spaces, meaning for numerical class v and \sigma \in Stab(X) (a stability condition) the set M_\sigma(v) of \sigma-semistable objects of type v is such a projective moduli space, then the quotient stack [X/G] has projective Bridgeland semistable moduli spaces as well. This is helpful to show projectivity in a range of cases, as well as making a step towards finding a general construction of such moduli spaces.