代数幾何学 · 2026-09-25

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

ベクトル束のmoduli stackとtwisted moduli空間に対するTorelli定理

Soham Ghosh, Ting Gong, Max Lieblich

soham-ghosh-1ijzyvc ting-gong-n2jvf6 max-lieblich-pc387j vector-bundles-sheaves moduli Vector bundles and sheaves ベクトル束・層 Moduli and families モジュライ・族

arXiv 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.

日本語要約

ベクトル束のmoduli stackまたはtwisted vector bundleのmoduli空間から基礎曲線を復元する複数のTorelli定理を示す。tame Deligne–Mumford曲線ではstack内部のlinearizationとtangent complexの行列式からPicard stackと主偏極を復元し、twistedの場合にはHitchin構成やKummer coverにより曲線とBrauer classを回収する。

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.

tame Deligne–Mumford曲線のベクトル束moduli stack、および滑らかな曲線上のtwisted vector bundleのmoduli空間から、元の曲線やtwisting classを復元するTorelli定理を証明する。Picard stack・偏極Jacobian torsor・twisted Hitchin構成を使い、任意の基礎体や複数のrank・degree範囲を扱う。

arXiv
2609.29011v1
初回投稿日
2026-09-24
原論文のライセンス
arXiv non-exclusive distribution license
掲載形式
Abstract・Introductionに基づく日本語要約
タグ
algebraic-geometry 2026

原論文を開く

書誌情報

要約

Torelli問題は、曲線に付随するmoduli空間から元の曲線を復元できるかを問う。本論文はstacky curveとtwisted vector bundleという二つの方向でこの問題を扱う。

第一部では任意の体上のtame Deligne–Mumford曲線について、rank $r\geq3$ のfixed-determinant moduli stackからcoarse curveを復元する。stackに内在するspecial-linear linearizationからPicard stackを、tangent complexの行列式から主偏極を抽出する。

第二部ではtwisted Hitchin構成、ordered Kummer cover、polarized Fano torsorなどを用い、曲線に加えてBrauer classやdeterminant gerbeを復元する。複数のrank・degree・標数条件を分けて精密な結論を与える。

背景と問題設定

通常のcoarse moduli spaceはstrictly semistable locusなどの情報を失い、低種数では曲線を区別できないことがある。moduli stackやtwisting dataを保持すれば、復元に必要な偏極情報を内在的に取り出せる可能性がある。

主結果

fixed-determinant stack（Theorem 1.1.1）

coarse genusが2以上の滑らかでproperな幾何学的連結tame Deligne–Mumford曲線について、rank $r\geq3$ のfull fixed-determinant moduli stackの同型はcoarse curveの同型を誘導する。

fixed-degree stack（Theorem 1.1.2）

明示されたPicard componentのcompatibility仮定の下で、fixed-rank fixed-degree moduli stackからもcoarse curveを復元できる。

twisted vector bundle（Theorem 1.2.1）

Introductionでは複数の場合に分けて述べられている。指定された標数・rank・degree条件の下でtwisted moduli空間の同型から曲線を復元し、追加の符号・中立性条件の下でtwisting classまたはBrauer classも復元する。

証明の見取り図

第一部はfaithful $\mathrm{SL}_{r-1}$-linearizationでPicard stackを再構成し、determinant of tangent complexesでcanonical principal polarizationを得て古典的Torelliへ帰着する。第二部はtwisted Hitchin discriminant、Kummer cover、relative Desale–Ramanan構成を場合ごとに使い分ける。

原論文との対応