Pub Date : 2024-01-01Epub Date: 2024-10-25DOI: 10.1007/s40879-024-00779-5
Charles F Doran, Alan Thompson
We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens-Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi-Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.
{"title":"The mirror Clemens-Schmid sequence.","authors":"Charles F Doran, Alan Thompson","doi":"10.1007/s40879-024-00779-5","DOIUrl":"https://doi.org/10.1007/s40879-024-00779-5","url":null,"abstract":"<p><p>We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens-Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi-Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.</p>","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"10 4","pages":"63"},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11511770/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142510025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-16DOI: 10.1007/s40879-023-00717-x
Zhen Zhang
{"title":"On simple-minded systems over domestic Brauer graph algebras","authors":"Zhen Zhang","doi":"10.1007/s40879-023-00717-x","DOIUrl":"https://doi.org/10.1007/s40879-023-00717-x","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"15 10","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138968089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1007/s40879-023-00711-3
O. Bergvall
{"title":"Arithmetic and topology of classical structures associated with plane quartics","authors":"O. Bergvall","doi":"10.1007/s40879-023-00711-3","DOIUrl":"https://doi.org/10.1007/s40879-023-00711-3","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"285 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138989748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-29DOI: 10.1007/s40879-023-00713-1
Joel Aguilar-Velázquez, Reynaldo Rojas-Hernández, Vladimir V. Tkachuk
{"title":"If K is a Valdivia compact space, then $$C_{hspace{-1.111pt}p}(K)$$ C p","authors":"Joel Aguilar-Velázquez, Reynaldo Rojas-Hernández, Vladimir V. Tkachuk","doi":"10.1007/s40879-023-00713-1","DOIUrl":"https://doi.org/10.1007/s40879-023-00713-1","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"13 1","pages":"1-13"},"PeriodicalIF":0.6,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139209422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-20DOI: 10.1007/s40879-023-00709-x
Jorge Mello
{"title":"On effective $$epsilon $$ ϵ -integrality in orbits of rational maps over function fields and multiplicative dependence","authors":"Jorge Mello","doi":"10.1007/s40879-023-00709-x","DOIUrl":"https://doi.org/10.1007/s40879-023-00709-x","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"121 1","pages":"1-21"},"PeriodicalIF":0.6,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139256796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-20DOI: 10.1007/s40879-023-00671-8
Jerson Pérez, Carlos Uzc'ategui
{"title":"On the Polishness of the inverse semigroup $$Gamma (X)$$ Γ ( X ) on a compact metric spa","authors":"Jerson Pérez, Carlos Uzc'ategui","doi":"10.1007/s40879-023-00671-8","DOIUrl":"https://doi.org/10.1007/s40879-023-00671-8","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"165 1","pages":"1-15"},"PeriodicalIF":0.6,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139257080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1007/s40879-023-00708-y
Mauricio Ramseyer, Oscar Salinas, Marisa Toschi
{"title":"Two-weight boundedness for local fractional maximal and applications","authors":"Mauricio Ramseyer, Oscar Salinas, Marisa Toschi","doi":"10.1007/s40879-023-00708-y","DOIUrl":"https://doi.org/10.1007/s40879-023-00708-y","url":null,"abstract":"","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":"32 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134901228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1007/s40879-023-00702-4
Ugo Bruzzo, William D. Montoya
Abstract We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs ( V , X ) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety $$mathbb {P}_{Sigma }^{2k+1}$$ PΣ2k+1 , with $$Vsubset X$$ V⊂X . The hypersurface X is supposed to be of Macaulay type , which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class $$lambda _Vin H^{k,k}(X)$$ λV∈Hk,k(X) deforms to an algebraic class if and only if it remains of type ( k , k ). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).
{"title":"Deformation of pairs and Noether–Lefschetz loci in toric varieties","authors":"Ugo Bruzzo, William D. Montoya","doi":"10.1007/s40879-023-00702-4","DOIUrl":"https://doi.org/10.1007/s40879-023-00702-4","url":null,"abstract":"Abstract We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs ( V , X ) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety $$mathbb {P}_{Sigma }^{2k+1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mi>Σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> , with $$Vsubset X$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> . The hypersurface X is supposed to be of Macaulay type , which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class $$lambda _Vin H^{k,k}(X)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>V</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> deforms to an algebraic class if and only if it remains of type ( k , k ). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).","PeriodicalId":44725,"journal":{"name":"European Journal of Mathematics","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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