In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.
{"title":"Extension of cohomology classes and holomorphic sections defined on subvarieties","authors":"Xiangyu Zhou, Langfeng Zhu","doi":"10.1090/JAG/766","DOIUrl":"https://doi.org/10.1090/JAG/766","url":null,"abstract":"In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general \u0000\u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 L^2\u0000 \u0000\u0000 extension theorem obtained by Demailly.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43104787","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 show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on P2mathbb {P}^2. This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on P2mathbb {P}^2, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local P2mathbb {P}^2.
As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on P2mathbb {P}^2 is Hodge-Tate, and we give the first non-trivial numerical checks of the general χchi-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local
我们证明了一个纯代数结构，即二维散射图，描述了在P2mathbb｛P｝^2上相干槽轮的导出范畴中Bridgeland半稳定对象的模空间的大部分壁交叉行为。这给出了一种新的算法，用于计算P 2 mathbb｛P｝^2上Gieseker半稳定槽轮的经典模空间的交叉上同调的Hodge数，或者等价于局部P 2 math bb｛P｝^2的紧支撑槽轮的精化Donaldson-Thomas不变量。作为应用，证明了P 2 mathbb｛P｝^2上的Gieseker半稳定槽的模空间的交上同调是Hodge-Tate，给出了局部P2mathbb｛P｝^2上一维槽轮的精化Donaldson-Thomas不变量的广义χ。
{"title":"Scattering diagrams, stability conditions, and coherent sheaves on ℙ²","authors":"Pierrick Bousseau","doi":"10.1090/jag/795","DOIUrl":"https://doi.org/10.1090/jag/795","url":null,"abstract":"<p>We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {P}^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {P}^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {P}^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>\u0000\u0000<p>As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">P</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {P}^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is Hodge-Tate, and we give the first non-trivial numerical checks of the general <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi\">\u0000 <mml:semantics>\u0000 <mml:mi>χ<!-- χ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">chi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathM","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45036037","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 the algebraic unrealizability of a certain isotopy type of plane affine real algebraic M M -sextic which is pseudoholomorphically realizable. This result completes the classification up to isotopy of real algebraic affine M M -sextics. The proof of this result given in a previous paper by the first two authors [J. Algebraic Geom. 11 (2002), pp. 293–310] was incorrect.
{"title":"Corrigendum to “A flexible affine 𝑀-sextic which is algebraically unrealizable”","authors":"S. F. Touzé, S. Orevkov, E. Shustin","doi":"10.1090/jag/733","DOIUrl":"https://doi.org/10.1090/jag/733","url":null,"abstract":"We prove the algebraic unrealizability of a certain isotopy type of plane affine real algebraic \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000-sextic which is pseudoholomorphically realizable. This result completes the classification up to isotopy of real algebraic affine \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000-sextics. The proof of this result given in a previous paper by the first two authors [J. Algebraic Geom. 11 (2002), pp. 293–310] was incorrect.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/733","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46065596","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 general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E E , where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E → P 1 Eto mathbb {P}^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.
{"title":"Enumerating pencils with moving ramification on curves","authors":"Carl Lian","doi":"10.1090/jag/776","DOIUrl":"https://doi.org/10.1090/jag/776","url":null,"abstract":"We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve \u0000\u0000 \u0000 E\u0000 E\u0000 \u0000\u0000, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps \u0000\u0000 \u0000 \u0000 E\u0000 →\u0000 \u0000 \u0000 P\u0000 \u0000 1\u0000 \u0000 \u0000 Eto mathbb {P}^1\u0000 \u0000\u0000 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43209329","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 construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at p p and deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.
{"title":"The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type","authors":"Paul Hamacher","doi":"10.1090/JAG/732","DOIUrl":"https://doi.org/10.1090/JAG/732","url":null,"abstract":"We construct a generalisation of Mantovan’s almost product structure to Shimura varieties of Hodge type with hyperspecial level structure at \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000 and deduce that the perfection of the Newton strata are proétale locally isomorphic to the perfection of the product of a central leaf and a Rapoport-Zink space. The almost product formula can be extended to obtain an analogue of Caraiani and Scholze’s generalisation of the almost product structure for Shimura varieties of Hodge type.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/732","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47307786","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}
Asher Auel, C. Böhning, H. G. Bothmer, Alena Pirutka
We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P2. We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.
{"title":"Conic bundle fourfolds with nontrivial unramified Brauer group","authors":"Asher Auel, C. Böhning, H. G. Bothmer, Alena Pirutka","doi":"10.1090/jag/743","DOIUrl":"https://doi.org/10.1090/jag/743","url":null,"abstract":"We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen–Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P2. We also prove the existence of universally CH0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/743","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46647036","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}
Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $mathbb R$-divisor $D_+$ for which $h^0(X,lfloor m D_+ rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.
设$X$是一个光滑的投影变量。除数$D$的Iitaka维数是一个重要的不变量，但它不仅取决于$D$的数值类。然而，有几种“数值Iitaka维”的定义，仅取决于数值类。在本文中，我们证明存在一个伪有效的$mathbb R$除数，其中这些不变量取不同的值。关键是构造一个伪有效的$mathbb R$-除数$D_+$的例子，其中$h^0(X，lfloor m D_+ rfloor+ a)$上下以$m^{3/2}$的倍数为界，对于任何足够充裕的$ a $。
{"title":"Notions of numerical Iitaka dimension do not coincide","authors":"John Lesieutre","doi":"10.1090/JAG/763","DOIUrl":"https://doi.org/10.1090/JAG/763","url":null,"abstract":"Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of ``numerical Iitaka dimension'', depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $mathbb R$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $mathbb R$-divisor $D_+$ for which $h^0(X,lfloor m D_+ rfloor+A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45264054","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 show that there exists a positive real number δ>0delta >0 such that for any normal quasi-projective Qmathbb {Q}-Gorenstein 33-fold XX, if XX has worse than canonical singularities, that is, the minimal log discrepancy of XX is less than 11, then the minimal log discrepancy of XX is not greater than 1−δ
我们证明了存在一个正实数δ>0δ>0，使得对于任何正规的拟投影Qmathbb｛Q｝-Gorenstein 3 3重X X X，如果X X具有比规范奇点更差的奇异性，即X X的最小对数偏差小于11，则X X的最小对数偏差不大于1−δ。作为应用，我们证明了所有非正则klt-Calabi–Yau 3-折叠的集合是有界模触发器，并且所有klt-Calobi–Yau3-折叠的全局索引是从上有界的。
{"title":"A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three","authors":"Chen Jiang","doi":"10.1090/jag/759","DOIUrl":"https://doi.org/10.1090/jag/759","url":null,"abstract":"<p>We show that there exists a positive real number <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">delta >0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> such that for any normal quasi-projective <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {Q}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Gorenstein <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\u0000 <mml:semantics>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-fold <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>, if <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> has worse than canonical singularities, that is, the minimal log discrepancy 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> is less than <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, then the minimal log discrepancy 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> is not greater than <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 minus delta\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"app","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45909253","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 expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation X X , the canonical map X a n → X X^{mathrm {an}} to X is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.
{"title":"GAGA theorems in derived complex geometry","authors":"Mauro Porta","doi":"10.1090/JAG/716","DOIUrl":"https://doi.org/10.1090/JAG/716","url":null,"abstract":"In this paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000, the canonical map \u0000\u0000 \u0000 \u0000 \u0000 X\u0000 \u0000 \u0000 a\u0000 n\u0000 \u0000 \u0000 \u0000 →\u0000 X\u0000 \u0000 X^{mathrm {an}} to X\u0000 \u0000\u0000 is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/716","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47412219","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 closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω ^ 2 hat {omega }^2 of the dualizing sheaf of a curve in terms of Zhang’s invariant φ varphi . As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.
{"title":"On arithmetic intersection numbers on self-products of curves","authors":"R. Wilms","doi":"10.1090/jag/777","DOIUrl":"https://doi.org/10.1090/jag/777","url":null,"abstract":"We give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number \u0000\u0000 \u0000 \u0000 \u0000 \u0000 ω\u0000 ^\u0000 \u0000 \u0000 2\u0000 \u0000 hat {omega }^2\u0000 \u0000\u0000 of the dualizing sheaf of a curve in terms of Zhang’s invariant \u0000\u0000 \u0000 φ\u0000 varphi\u0000 \u0000\u0000. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2019-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45289400","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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