Torelli theorems for moduli stacks and twisted moduli spaces of vector bundles

Source abstract

We prove Torelli theorems for moduli of vector bundles in two complementary settings. In the first part, over an arbitrary field, we recover the coarse curve of a smooth, proper, geometrically connected, tame Deligne--Mumford curve of coarse genus at least two from its full fixed-determinant moduli stack in rank at least three. We also prove a fixed-degree version under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. In the second part, we study moduli spaces of twisted vector bundles on smooth projective curves. A twisted Hitchin construction recovers the curve in specified characteristic and degree ranges, with separate hypotheses for recovery of the twisting class. For rank two with trivial determinant, we recover the curve and the Brauer class from the moduli stack through an ordered Kummer cover and its polarized Jacobian torsor. For odd determinant on hyperelliptic curves in characteristic different from two, residual gerbes give a relative Desale--Ramanan construction without an actual determinant line bundle over the ground field. An intrinsic polarized Fano torsor recovers the curve and the class, and a corrected polarization obstruction recovers the determinant gerbe. We also give a twisted Newstead construction in genus two, with an explicit descent condition for its converse, and period--index applications.