The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities x ∈ ( X , Δ ) xin (X,Delta ) satisfies the ACC if the coefficients of Δ Delta belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of δ delta -plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
{"title":"ACC for local volumes and boundedness of singularities","authors":"Jingjun Han, Yuchen Liu, Lu Qi","doi":"10.1090/jag/799","DOIUrl":"https://doi.org/10.1090/jag/799","url":null,"abstract":"The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities \u0000\u0000 \u0000 \u0000 x\u0000 ∈\u0000 (\u0000 X\u0000 ,\u0000 Δ\u0000 )\u0000 \u0000 xin (X,Delta )\u0000 \u0000\u0000 satisfies the ACC if the coefficients of \u0000\u0000 \u0000 Δ\u0000 Delta\u0000 \u0000\u0000 belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of \u0000\u0000 \u0000 δ\u0000 delta\u0000 \u0000\u0000-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48672119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that every smooth affine variety of dimension d d embeds into every simple algebraic group of dimension at least 2 d + 2 2d+2 . We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to 2 d + 1 2d+1 . In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.
{"title":"Existence of embeddings of smooth varieties into linear algebraic groups","authors":"P. Feller, Immanuel van Santen","doi":"10.1090/jag/793","DOIUrl":"https://doi.org/10.1090/jag/793","url":null,"abstract":"We prove that every smooth affine variety of dimension \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000 embeds into every simple algebraic group of dimension at least \u0000\u0000 \u0000 \u0000 2\u0000 d\u0000 +\u0000 2\u0000 \u0000 2d+2\u0000 \u0000\u0000. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to \u0000\u0000 \u0000 \u0000 2\u0000 d\u0000 +\u0000 1\u0000 \u0000 2d+1\u0000 \u0000\u0000.\u0000\u0000In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41731907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a fixed point theorem for the action of certain local monodromy groups on étale covers and use it to deduce lower bounds on essential dimension. In particular, we give more geometric proofs of some of the results of a paper of Farb, Kisin and Wolfson, which uses arithmetic methods to prove incompressibility results for Shimura varieties and moduli spaces of curves. Our method allows us to prove new results for exceptional groups, applies also to the reduction modulo good primes of congruence covers of Shimura varieties and moduli spaces of curves, and also to certain “quantum” covers of moduli spaces of curves arising from a certain TQFT.
{"title":"Fixed points, local monodromy, and incompressibility of congruence covers","authors":"P. Brosnan, N. Fakhruddin","doi":"10.1090/jag/800","DOIUrl":"https://doi.org/10.1090/jag/800","url":null,"abstract":"We prove a fixed point theorem for the action of certain local monodromy groups on étale covers and use it to deduce lower bounds on essential dimension. In particular, we give more geometric proofs of some of the results of a paper of Farb, Kisin and Wolfson, which uses arithmetic methods to prove incompressibility results for Shimura varieties and moduli spaces of curves. Our method allows us to prove new results for exceptional groups, applies also to the reduction modulo good primes of congruence covers of Shimura varieties and moduli spaces of curves, and also to certain “quantum” covers of moduli spaces of curves arising from a certain TQFT.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46220786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer’s flops.
{"title":"Non-commutative deformations of perverse coherent sheaves and rational curves","authors":"Y. Kawamata","doi":"10.1090/jag/805","DOIUrl":"https://doi.org/10.1090/jag/805","url":null,"abstract":"We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer’s flops.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41778932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by Q mathbb {Q} -Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.
{"title":"Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor","authors":"D. Greb, Stefan Kebekus, T. Peternell","doi":"10.1090/jag/785","DOIUrl":"https://doi.org/10.1090/jag/785","url":null,"abstract":"We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by \u0000\u0000 \u0000 \u0000 Q\u0000 \u0000 mathbb {Q}\u0000 \u0000\u0000-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47943497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs","authors":"Tim Graefnitz","doi":"10.1090/jag/794","DOIUrl":"https://doi.org/10.1090/jag/794","url":null,"abstract":"<p>Consider a log Calabi-Yau pair <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> consisting of a smooth del Pezzo surface <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"greater-than-or-equal-to 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">geq 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and a smooth anticanonical divisor <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> intersecting <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper D right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(X,D)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generati","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48480911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>We prove that the ramified Prym map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript g comma r"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">mathcal P_{g, r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which sends a covering <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper D long right-arrow upper C"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mo>:</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">⟶<!-- ⟶ --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">pi :Dlongrightarrow C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ramified in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> points to the Prym variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis pi right-parenthesis colon-equal upper K e r left-parenthesis upper N m Subscript pi Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π<!-- π --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≔</mml:mo> <mml:mi>K</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P(pi )≔Ker(Nm_{pi })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an embedding for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than-or-equal-to 6"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">rge 6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis upper C ri
我们证明了分支Prym映射P g, r mathcal p＿{G, r} 它发出一个覆盖π: D pi : dlongrightarrow C在r中的分支r指向P(π)，其中K e r (N m π) P(pi )对象是Ker(Nm＿{pi })是对所有r≥6r的嵌入ge 对于所有g(C)>0 g(C)>0。此外，通过研究超椭圆曲线覆盖轨迹的限制，我们证明了P g, 2 mathcal p＿{G, 2} P g, 4 mathcal p＿{G, 4} 有正维纤维。
{"title":"Global Prym-Torelli for double coverings ramified in at least six points","authors":"J. Naranjo, –. Ortega","doi":"10.1090/jag/779","DOIUrl":"https://doi.org/10.1090/jag/779","url":null,"abstract":"<p>We prove that the ramified Prym map <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper P Subscript g comma r\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>r</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal P_{g, r}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which sends a covering <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi colon upper D long right-arrow upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">⟶<!-- ⟶ --></mml:mo>\u0000 <mml:mi>C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi :Dlongrightarrow C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> ramified in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\">\u0000 <mml:semantics>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">r</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> points to the Prym variety <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P left-parenthesis pi right-parenthesis colon-equal upper K e r left-parenthesis upper N m Subscript pi Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>≔</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">P(pi )≔Ker(Nm_{pi })</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an embedding for all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r greater-than-or-equal-to 6\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>6</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">rge 6</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and for all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g left-parenthesis upper C ri","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45359188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the relative anti-canonical divisor − K X / Y -K_{X/Y} of an algebraic fiber space ϕ : X → Y phi colon Xto Y , and we reveal relations among positivity conditions of − K X / Y -K_{X/Y} , certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of Y Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.
{"title":"On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces","authors":"Sho Ejiri, M. Iwai, Shin-ichi Matsumura","doi":"10.1090/jag/814","DOIUrl":"https://doi.org/10.1090/jag/814","url":null,"abstract":"In this paper, we study the relative anti-canonical divisor \u0000\u0000 \u0000 \u0000 −\u0000 \u0000 K\u0000 \u0000 X\u0000 \u0000 /\u0000 \u0000 Y\u0000 \u0000 \u0000 \u0000 -K_{X/Y}\u0000 \u0000\u0000 of an algebraic fiber space \u0000\u0000 \u0000 \u0000 ϕ\u0000 :\u0000 X\u0000 →\u0000 Y\u0000 \u0000 phi colon Xto Y\u0000 \u0000\u0000, and we reveal relations among positivity conditions of \u0000\u0000 \u0000 \u0000 −\u0000 \u0000 K\u0000 \u0000 X\u0000 \u0000 /\u0000 \u0000 Y\u0000 \u0000 \u0000 \u0000 -K_{X/Y}\u0000 \u0000\u0000, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of \u0000\u0000 \u0000 Y\u0000 Y\u0000 \u0000\u0000. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45585372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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