模栈与向量丛扭曲模空间的Torelli定理
Torelli theorems for moduli stacks and twisted moduli spaces of vector bundles
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中文总结 AI 辅助
本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。
中文摘要 AI 辅助
我们在两个互补的背景下证明了向量丛模空间的Torelli定理。在第一部分中，在任意域上，我们从粗亏格至少为二的平滑、正常、几何连通、驯服的Deligne--Mumford曲线的完全固定行列式模栈（秩至少为三）中恢复其粗曲线。我们还证明了在显式相容性假设下的固定次数版本。该方法本质上是模栈的：忠实的特殊线性线性化恢复Picard栈，切复形的行列式恢复典范主极化。在第二部分中，我们研究了光滑射影曲线上扭曲向量丛的模空间。一个扭曲的Hitchin构造在指定的特征和次数范围内恢复曲线，并有恢复扭曲类的独立假设。对于秩二且行列式平凡的情况，我们通过有序Kummer覆盖及其极化Jacobian torsor从模栈恢复曲线和Brauer类。对于特征不等于二的超椭圆曲线上奇行列式的情况，剩余gerbes给出了一个相对Desale--Ramanan构造，而无需在基域上存在实际的行列式线丛。一个内在的极化Fano torsor恢复曲线和类，一个修正的极化障碍恢复行列式gerbe。我们还给出了亏格二中的扭曲Newstead构造，并给出了其逆的显式下降条件，以及周期--指数应用。
英文摘要
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.
