[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-140252-105":59,"doc-detail-140252-en":125},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":118,"head_meta":120,"extra_data":122,"updated_unix":124},105,"en","relative-and-absolute-lefschetz-standard-conjectures-for-some-lagrangian-fibrations","Relative and Absolute Lefschetz Standard Conjectures for Some Lagrangian Fibrations","","The paper proves that the hyper-Kähler varieties of OG10 type constructed by Laza–Saccà–Voisin satisfy the Lefschetz standard conjecture. The result follows from a more general theorem for certain Lagrangian fibrations, whose key technical hypothesis requires the fibration to fulfill the conditions of Ngô’s support theorem. Establishing that these OG10 tenfolds meet those conditions is addressed directly. The work also introduces a relative version of the Lefschetz standard conjecture and explains its relation to the classical absolute formulation.",{"@graph":69,"@context":117},[70,84,100],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/relative-and-absolute-lefschetz-standard-conjectures-for-some-lagrangian-fibrations/140252/",{"url":83,"name":65,"@type":85,"author":86,"headline":65,"publisher":89,"fileFormat":92,"inLanguage":63,"description":67,"dateModified":93,"datePublished":94,"encodingFormat":92,"isAccessibleForFree":95,"interactionStatistic":96},"DigitalDocument",{"name":87,"@type":88},"Mali","Person",{"url":74,"name":90,"@type":91},"DocShare","Organization","application/pdf","2026-09-04","2026-08-24",true,{"@type":97,"interactionType":98,"userInteractionCount":19},"InteractionCounter",{"@type":99},"ViewAction",{"@type":101,"mainEntity":102},"FAQPage",[103,109,113],{"name":104,"@type":105,"acceptedAnswer":106},"What main result does the paper establish about OG10-type hyper-Kähler varieties?","Question",{"text":107,"@type":108},"It shows that the OG10-type hyper-Kähler varieties constructed by Laza–Saccà–Voisin verify the Lefschetz standard conjecture.","Answer",{"name":110,"@type":105,"acceptedAnswer":111},"How does the paper reduce the problem to a more general statement for Lagrangian fibrations?",{"text":112,"@type":108},"It proves a general result for certain Lagrangian fibrations and then applies it to the OG10-type examples.",{"name":114,"@type":105,"acceptedAnswer":115},"What technical ingredient is essential for the general theorem to apply?",{"text":116,"@type":108},"The Lagrangian fibration must satisfy the hypotheses of Ngô’s support theorem.","https://schema.org",{"og:url":83,"og:type":119,"og:title":65,"og:site_name":90,"og:description":67},"article",{"robots":121,"canonical":83},"index,follow",{"doc_id":123,"site_id":62},140252,1787571318,{"code":4,"msg":5,"data":126},{"doc_id":123,"user_id":127,"nickname":87,"user_avatar":128,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":129,"file_id":130,"file_url":131,"file_type":132,"file_size":133,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":134,"language":135,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":67,"update_tm":124,"read_time":139},2336475104362,"https://ap-avatar.wpscdn.com/avatar/22000c4c46a41b752dd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786595829695023868","RELATIVE AND ABSOLUTE LEFSCHETZ STANDARD CONJECTURES FOR SOME LAGRANGIAN FIBRATIONS  \narXiv :2304 .00978v2 [math .AG] 16 Nov 2025  \nGIUSEPPE ANCONA, MATTIA CAVICCHI, ROBERT LATERVEER, AND GIULIA SACCÀ  \nAbstract. We show that the hyper-Kähler varieties of OG10-type constructed by Laza–Saccà–Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this conjecture. The main technical assumption of this general result is that the Lagrangian fibration satisfies the hypotheses of Ngô’s support theorem. Verifying that the LSV tenfolds do satisfy those hypotheses is of independent interest. Another point of independent interest of the paper is the definition and the study of the Lefschetz standard conjecture in the relative  \nsetting, and its relation to the classical absolute case.  \nContents  \n1. Introduction 2  \nOrganization of the paper 5  \nAcknowledgments 5  \n2. Conventions 5  \n3. Statement of the main result 7  \n4. The relative Lefschetz standard conjecture 8  \n5. Good decomposition: from relative to absolute 11  \n6. Ngô fibrations and relative Lefschetz 12  \n7. The Shen–Yin decomposition is good 14  \n8. Voisin’s trick and the end of the proof 15  \n9. An application to LSV tenfolds 17  \n9.1. LSV tenfolds and their Ngô fibration 17  \n9.2. Polarizability of the descended relative Prym 19  \n9.3. Application of Theorem 3.1 to LSV tenfolds 24  \nReferences 25  \n2020 Mathematics Subject Classification: 14C15, 14C25, 14C30  \nGA and MC are supported by ANR grant ANR–18–CE40–0017 . RL is supported by ANR grant ANR–20–CE40–0023 . GS acknowledges support from NSF CAREER grant DMS-2144483 and NSF FRG grant DMS-2052750 .  \nDate: November 18, 2025 .  \n2 GIUSEPPE ANCONA, MATTIA CAVICCHI, ROBERT LATERVEER, AND GIULIA SACCÀ  \n1. Introduction  \nThe standard conjectures, formulated by Grothendieck and Bombieri in the sixties, have been hugely influential in the development of the theory of motives and algebraic cycles [Gro69, Kle68 , Kle94] . Yet, in spite of their pervasive presence 1 , not a great deal more is known about them nowadays than in the sixties.  \nIn this paper we will work over the complex numbers. In this setting it is known that the Lefschetz standard conjecture implies all the other standard conjectures.  \nLet us recall its formulation. Let L be an ample divisor on a smooth projective variety X of dimension dX . The hard Lefschetz theorem says that the map (1 . 1) ∪Ln : HdX −n(X, Q)  HdX +n(X, Q)  \nis an isomorphism. The Lefschetz standard conjecture predicts that the inverse of this map is induced by an algebraic correspondence. This project is guided by the basic question: how well does this conjecture behave under fibration?  \nThe starting point of this note is Voisin’s paper [Voi22], where the Lefschetz standard conjecture in degree 2 is proved for HK manifolds which are covered by Lagrangian subvarieties. Particularly relevant for our paper is the case of Lagrangian fibrations: in this case, Voisin proves the conjecture in degree two, under the assumption that the SYZ conjecture holds. Here, we use Voisin’s idea, extending it thanks to the use of Ngô’s Support Theorem.  \nTo explain how this theorem enters the picture, let us assume that X is endowed with a surjective morphism f : X → B to a smooth projective variety B (with f non-constant and non-finite) . We ask ourselves how much the Lefschetz standard conjecture for B and for the smooth fibers may help to show the Lefschetz standard conjecture for X .  \nLet us be more precise and let us read this through the decomposition theorem. To ease notation, we will work with the constant complex CX = Q [dX ] on X and write Hn (X) = H(X, CX ) = HdX +n(X, Q) . Moreover, let η be a relatively ample divisor on X , LB an ample divisor on B and β its pullback on X . The decomposition theorem gives in particular a decomposition  \n(1.2) Hn (X) = M Ha (B,pRbf∗ CX ) ,  \na+b=n  \nwhere the pRbf∗ CX","cbCaipGS9t458e9K","https://ap.wps.com/l/cbCaipGS9t458e9K","pdf",641737,27,"English","# Introduction\n## Organization of the paper\n## Conventions\n## Statement of the main result\n## The relative Lefschetz standard conjecture\n## Good decomposition: from relative to absolute\n## Ngô fibrations and relative Lefschetz\n## The Shen–Yin decomposition is good\n## Voisin’s trick and the end of the proof\n## An application to LSV tenfolds\n## References","[{\"question\":\"What main result does the paper establish about OG10-type hyper-Kähler varieties?\",\"answer\":\"It shows that the OG10-type hyper-Kähler varieties constructed by Laza–Saccà–Voisin verify the Lefschetz standard conjecture.\"},{\"question\":\"How does the paper reduce the problem to a more general statement for Lagrangian fibrations?\",\"answer\":\"It proves a general result for certain Lagrangian fibrations and then applies it to the OG10-type examples.\"},{\"question\":\"What technical ingredient is essential for the general theorem to apply?\",\"answer\":\"The Lagrangian fibration must satisfy the hypotheses of Ngô’s support theorem.\"}]","Relative and Absolute Lefschetz Standard Conjectures for Some Lagrangian Fibrations | PDF",68]