We develop a framework to construct moduli spaces of Q {mathbb {Q}} -Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of Q {mathbb {Q}} -stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than 1 2 frac {1}{2} . Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.
{"title":"Moduli of ℚ-Gorenstein pairs and applications","authors":"Stefano Filipazzi, Giovanni Inchiostro","doi":"10.1090/jag/823","DOIUrl":"https://doi.org/10.1090/jag/823","url":null,"abstract":"We develop a framework to construct moduli spaces of \u0000\u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 {mathbb {Q}}\u0000 \u0000\u0000-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of \u0000\u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 {mathbb {Q}}\u0000 \u0000\u0000-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than \u0000\u0000 \u0000 \u0000 1\u0000 2\u0000 \u0000 frac {1}{2}\u0000 \u0000\u0000. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"77 6","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452253","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 splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.
{"title":"Splitting of Gromov–Witten invariants with toric gluing strata","authors":"Yixian Wu","doi":"10.1090/jag/826","DOIUrl":"https://doi.org/10.1090/jag/826","url":null,"abstract":"We prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"15 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993320","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}
Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a kk-rational isolated singularity is kk-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the kk-Du Bois and kk-rational singularities in terms of standard invariants of singularities. In particular, we show that kk-Du Bois singularities are (k−1)(k-1)-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.
高有理数和高杜波依斯奇点最近被引入作为有理数和杜波依斯奇点标准定义的自然推广。在这篇笔记中，我们讨论了孤立奇点的这些性质，特别是在局部完全交集(lci)情况下。首先，我们在没有任何lci假设的情况下，证明了kk -有理孤立奇点是kk -Du Bois的事实。对于孤立的lci奇点，我们用奇点的标准不变量给出了k k -杜波依斯奇点和k k -有理奇点的完整刻画。特别地，我们证明了k k -Du Bois奇点对于孤立的lci奇点是(k−1)(k-1) -有理的。在证明过程中，我们建立了孤立lci奇点不变量之间的一些新关系，并证明了其中许多不变量是消失的。该方法还能快速证明孤立lci情况下附加定理的反演。最后，我们讨论了一些特定于超曲面情况的结果。
{"title":"The higher Du Bois and higher rational properties for isolated singularities","authors":"Robert Friedman, Radu Laza","doi":"10.1090/jag/824","DOIUrl":"https://doi.org/10.1090/jag/824","url":null,"abstract":"Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational isolated singularity is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational singularities in terms of standard invariants of singularities. In particular, we show that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois singularities are <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis k minus 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(k-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192781","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 algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic R{mathbb {R}}-Cartier divisor D¯overline {D} are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized R{mathbb {R}}-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of D¯overline {D} coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.
{"title":"Arithmetic Okounkov bodies and positivity of adelic Cartier divisors","authors":"François Ballaÿ","doi":"10.1090/jag/821","DOIUrl":"https://doi.org/10.1090/jag/821","url":null,"abstract":"In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cartier divisor <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135993452","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 introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2 2 -form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.
{"title":"Refined count of oriented real rational curves","authors":"Thomas Blomme","doi":"10.1090/jag/801","DOIUrl":"https://doi.org/10.1090/jag/801","url":null,"abstract":"We introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000-form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43704449","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 show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-connectedness of a large class of varieties over a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be characterized as the condition that their generic point can be connected to a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational point using (not necessarily naive) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopies. We also show that symmetric powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="applicatio
{"title":"Geometric criteria for 𝔸¹-connectedness and applications to norm varieties","authors":"Chetan T. Balwe, A. Hogadi, Anand Sawant","doi":"10.1090/jag/790","DOIUrl":"https://doi.org/10.1090/jag/790","url":null,"abstract":"<p>We show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-connectedness of a large class of varieties over a field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> can be characterized as the condition that their generic point can be connected to a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational point using (not necessarily naive) <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-homotopies. We also show that symmetric powers of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) are <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"applicatio","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42482940","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 to test geometric properties such as Whitney equisingularity and Thom’s a f a_f condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
{"title":"Nondegenerate locally tame complete intersection varieties and geometry of nonisolated hypersurface singularities","authors":"C. Eyral, M. Oka","doi":"10.1090/jag/784","DOIUrl":"https://doi.org/10.1090/jag/784","url":null,"abstract":"We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s \u0000\u0000 \u0000 \u0000 a\u0000 f\u0000 \u0000 a_f\u0000 \u0000\u0000 condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41639855","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>This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The first, which we call <italic>tautological foliations</italic>, are defined on Hilbert modular varieties, and lift to characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The second, the <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-foliations</italic>, are defined on unitary Shimura varieties in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-closed, and the locus where they are smooth. Where not smooth, we construct a <italic>successive blowup</italic> of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semant
本文是de Shalit和Goren 2018年论文的延续。我们在特征p p上研究了下村品种S S上两种类型的叶理。第一个，我们称之为重言叶理，是在希尔伯特模变种上定义的，并提升到特征0。第二个，V-叶理，仅在特征p p p中定义在酉Shimura变种上，并推广了我们以前研究的叶理，当所讨论的CM场是二次虚时。我们确定这些叶理何时是p-p-closed的，以及它们是光滑的轨迹。在不光滑的地方，我们构建了下村品种的连续放大，它们作为光滑的叶理延伸到下村品种。我们讨论了叶理的一些整体变化。我们将S S与叶理的商与一个纯粹不可分割的映射联系起来，该映射来自另一个相同类型的下村品种的某个组成部分，在p p处具有准水平结构，到S。S
{"title":"Foliations on Shimura varieties in positive characteristic","authors":"E. Goren, E. D. Shalit","doi":"10.1090/jag/820","DOIUrl":"https://doi.org/10.1090/jag/820","url":null,"abstract":"<p>This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\u0000 <mml:semantics>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The first, which we call <italic>tautological foliations</italic>, are defined on Hilbert modular varieties, and lift to characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The second, the <italic><inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper V\">\u0000 <mml:semantics>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">V</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-foliations</italic>, are defined on unitary Shimura varieties in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-closed, and the locus where they are smooth. Where not smooth, we construct a <italic>successive blowup</italic> of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\">\u0000 <mml:semantics>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">S</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semant","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49625211","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 the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi–Yau hypersurfaces over a punctured disk D ∖ { 0 } D setminus left { 0right } . It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato–Usui on the whole disk D D . By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross–Siebert program.
我们考虑了在穿孔盘D {0} D setminus left {0right }上由一组环形Calabi-Yau超曲面定义的Iritani的Hodge结构的残差b模型变分。它自然地被推广到加藤-臼井极化Hodge结构在整个圆盘上的对数变化。将其限制在原点上，得到了标准对数点上的极化对数霍奇结构(PLH)。本文用Gross-Siebert规划中圆环退化对偶交复合体的积分仿射结构来描述PLH。
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