{"title":"Locally trivial monodromy of moduli spaces of sheaves on K3 surfaces","authors":"Claudio Onorati, Arvid Perego, Antonio Rapagnetta","doi":"10.1090/tran/9185","DOIUrl":null,"url":null,"abstract":"<p>In this paper we study monodromy operators on moduli spaces <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M Subscript v Baseline left-parenthesis upper S comma upper H right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">M_v(S,H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of sheaves on K3 surfaces with non-primitive Mukai vectors <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding=\"application/x-tex\">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If we write <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v equals m w\"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mi>w</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">v=mw</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">m>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w\"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=\"application/x-tex\">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> primitive, then our main result is that the inclusion <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M Subscript w Baseline left-parenthesis upper S comma upper H right-parenthesis right-arrow upper M Subscript v Baseline left-parenthesis upper S comma upper H right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">→</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">M_w(S,H)\\to M_v(S,H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9185","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we study monodromy operators on moduli spaces Mv(S,H)M_v(S,H) of sheaves on K3 surfaces with non-primitive Mukai vectors vv. If we write v=mwv=mw, with m>1m>1 and ww primitive, then our main result is that the inclusion Mw(S,H)→Mv(S,H)M_w(S,H)\to M_v(S,H) as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman.
本文研究了 K3 曲面上具有非原始穆凯向量 v v 的剪切的模空间 M v ( S , H ) M_v(S,H)上的单旋转算子。如果我们把 v = m w v=mw 写为 m > 1 m>1 而 w w 原始，那么我们的主要结果是，包含 M w ( S , H ) → M v ( S , H ) M_w(S,H)\to M_v(S,H)作为最奇异位点诱导了这些交映变体的单旋转群之间的同构，从而使我们能够把马克曼的一个结果推广到非原始情况。
期刊介绍：
All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are.
This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.
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