Pub Date : 2023-11-09DOI: 10.1007/s00229-023-01518-y
Adrian Langer
Abstract We show that under some assumptions on the monodromy group some combinations of higher Chern classes of a flat vector bundle are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch’s conjecture on Chern classes of flat vector bundles on smooth complex projective varieties but in some cases they give a more precise information. We also study Higgs version of Bloch’s conjecture and analogous problems in the positive characteristic case.
{"title":"On algebraic Chern classes of flat vector bundles","authors":"Adrian Langer","doi":"10.1007/s00229-023-01518-y","DOIUrl":"https://doi.org/10.1007/s00229-023-01518-y","url":null,"abstract":"Abstract We show that under some assumptions on the monodromy group some combinations of higher Chern classes of a flat vector bundle are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch’s conjecture on Chern classes of flat vector bundles on smooth complex projective varieties but in some cases they give a more precise information. We also study Higgs version of Bloch’s conjecture and analogous problems in the positive characteristic case.","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":" 90","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-06DOI: 10.1007/s00229-023-01514-2
Ciro Ciliberto
Abstract In this paper we study the rationality problem for Fano threefolds $$Xsubset {mathbb P}^{p+1}$$ X⊂Pp+1 of genus p , that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$pgeqslant 8$$ p⩾8 (this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$pgeqslant 7$$ p⩾7 containing a plane is rational; (3) any Fano threefold of genus $$pgeqslant 17$$ p⩾17 is rational; (4) a Fano threefold of genus $$pgeqslant 12$$ p⩾12 containing an ordinary line $$ell $$ ℓ in its smooth locus is rational.
{"title":"On the rationality of certain Fano threefolds","authors":"Ciro Ciliberto","doi":"10.1007/s00229-023-01514-2","DOIUrl":"https://doi.org/10.1007/s00229-023-01514-2","url":null,"abstract":"Abstract In this paper we study the rationality problem for Fano threefolds $$Xsubset {mathbb P}^{p+1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of genus p , that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$pgeqslant 8$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> (this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$pgeqslant 7$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> containing a plane is rational; (3) any Fano threefold of genus $$pgeqslant 17$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>17</mml:mn> </mml:mrow> </mml:math> is rational; (4) a Fano threefold of genus $$pgeqslant 12$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:math> containing an ordinary line $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> in its smooth locus is rational.","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"32 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135584251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-30DOI: 10.1007/s00229-023-01517-z
Hiromu Tanaka
Abstract For a regular del Pezzo surface X , we prove that $$|-12K_X|$$ |-12KX| is very ample. Furthermore, we also give an explicit upper bound for the volume $$K_X^2$$ KX2 which depends only on $$[k: k^p]$$ [k:kp] for the base field k . As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.
摘要对于正则del Pezzo曲面X，证明了$$|-12K_X|$$ | - 12 K X |是非常充足的。此外，我们还给出了体积$$K_X^2$$ K x2的显式上界，该上界仅取决于基础场K的$$[k: k^p]$$ [K: kp]。由此得到几何积分正则del Pezzo曲面的有界性。
{"title":"Boundedness of regular del Pezzo surfaces over imperfect fields","authors":"Hiromu Tanaka","doi":"10.1007/s00229-023-01517-z","DOIUrl":"https://doi.org/10.1007/s00229-023-01517-z","url":null,"abstract":"Abstract For a regular del Pezzo surface X , we prove that $$|-12K_X|$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>-</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is very ample. Furthermore, we also give an explicit upper bound for the volume $$K_X^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> which depends only on $$[k: k^p]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> for the base field k . As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136067570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-28DOI: 10.1007/s00229-023-01515-1
Qingfeng Sun
{"title":"Bounds for GL$$_2$$ $$times $$ GL$$_2$$ L-functions in the depth aspect","authors":"Qingfeng Sun","doi":"10.1007/s00229-023-01515-1","DOIUrl":"https://doi.org/10.1007/s00229-023-01515-1","url":null,"abstract":"","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"30 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s00229-023-01516-0
Zilong He
{"title":"On classic n-universal quadratic forms over dyadic local fields","authors":"Zilong He","doi":"10.1007/s00229-023-01516-0","DOIUrl":"https://doi.org/10.1007/s00229-023-01516-0","url":null,"abstract":"","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"46 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135316036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-24DOI: 10.1007/s00229-023-01510-6
Claudio Dappiaggi, Paolo Rinaldi, Federico Sclavi
Abstract A key result in distribution theory is Young’s product theorem which states that the product between two Hölder distributions $$uin mathcal {C}^alpha (mathbb {R}^d)$$ u∈Cα(Rd) and $$vin mathcal {C}^beta (mathbb {R}^d)$$ v∈Cβ(Rd) can be unambiguously defined if $$alpha +beta >0$$ α+β>0 . We revisit the problem of multiplying two Hölder distributions from the viewpoint of microlocal analysis, using techniques proper of Sobolev wavefront set. This allows us to establish sufficient conditions which allow the multiplication of two Hölder distributions even when $$alpha +beta le 0$$ α+β≤0 .
{"title":"On a microlocal version of Young’s product theorem","authors":"Claudio Dappiaggi, Paolo Rinaldi, Federico Sclavi","doi":"10.1007/s00229-023-01510-6","DOIUrl":"https://doi.org/10.1007/s00229-023-01510-6","url":null,"abstract":"Abstract A key result in distribution theory is Young’s product theorem which states that the product between two Hölder distributions $$uin mathcal {C}^alpha (mathbb {R}^d)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$vin mathcal {C}^beta (mathbb {R}^d)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>β</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> can be unambiguously defined if $$alpha +beta >0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We revisit the problem of multiplying two Hölder distributions from the viewpoint of microlocal analysis, using techniques proper of Sobolev wavefront set. This allows us to establish sufficient conditions which allow the multiplication of two Hölder distributions even when $$alpha +beta le 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> .","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135924789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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