Suppose that XX is a projective manifold whose tangent bundle TXT_X contains a locally free strictly nef subsheaf. We prove that XX is isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group π1(X)pi _1(X) is virtually solvable, then XX is isomorphic to a projective space.
设X X是一个射影流形，其切束T X T_X包含一个局部自由的严格nef子轴。证明X X与一般型双曲流形上的射影空间或射影束同构。此外，如果基本群π 1(X) pi _1(X)是虚可解的，则X X是射影空间同构的。
{"title":"Projective manifolds whose tangent bundle contains a strictly nef subsheaf","authors":"Jie Liu, Wenhao Ou, Xiaokui Yang","doi":"10.1090/jag/807","DOIUrl":"https://doi.org/10.1090/jag/807","url":null,"abstract":"<p>Suppose that <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 a projective manifold whose tangent bundle <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Subscript upper X\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">T_X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> contains a locally free strictly nef subsheaf. We prove that <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 isomorphic to either a projective space or a projective bundle over a hyperbolic manifold of general type. Moreover, if the fundamental group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi 1 left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi _1(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is virtually solvable, then <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 isomorphic to a projective space.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43641826","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 the Brill-Noether locus M18,163M^3_{18,16} is an irreducible component of the Gieseker-Petri locus in genus 1818 having codimension 22 in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.
{"title":"A codimension 2 component of the Gieseker-Petri locus","authors":"Margherita Lelli–Chiesa","doi":"10.1090/jag/780","DOIUrl":"https://doi.org/10.1090/jag/780","url":null,"abstract":"<p>We show that the Brill-Noether locus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M Subscript 18 comma 16 Superscript 3\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>18</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>16</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">M^3_{18,16}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an irreducible component of the Gieseker-Petri locus in genus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"18\">\u0000 <mml:semantics>\u0000 <mml:mn>18</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">18</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> having codimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47940720","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>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth, projective, rationally connected variety, defined over a number 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>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z subset-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Zsubset X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed subset of codimension at least two. In this paper, for certain choices of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we prove that the set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-integral points is potentially Zariski dense, in the sense that there is a finite extension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <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> such that the set of points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P element-of upper X left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Pin X(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are <inline-formula content-type="math/mathml"> <mml:math x
设X X是定义在数域k k上的光滑的、射影的、合理连通的变种，并设Z∧X Z子集X是余维至少为2的闭子集。在本文中，对于X X的某些选择，我们证明了Z Z积分点的集合是潜在的Zariski稠密的，即K K的有限扩展K K使得点P∈X(K) P In X(K)是Z Z积分的集合P∈X(K) P In X(X)是Zariski稠密的。这对哈塞特和茨钦克尔2001年提出的一个问题给出了肯定的答案。
{"title":"Codimension two integral points on some rationally connected threefolds are potentially dense","authors":"David McKinnon, Mike Roth","doi":"10.1090/jag/782","DOIUrl":"https://doi.org/10.1090/jag/782","url":null,"abstract":"<p>Let <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> be a smooth, projective, rationally connected variety, defined over a number 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>, and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z subset-of upper X\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Zsubset X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a closed subset of codimension at least two. In this paper, for certain choices 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>, we prove that the set of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z\">\u0000 <mml:semantics>\u0000 <mml:mi>Z</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Z</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-integral points is potentially Zariski dense, in the sense that there is a finite extension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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> of <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> such that the set of points <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P element-of upper X left-parenthesis upper K right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Pin X(K)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that are <inline-formula content-type=\"math/mathml\">\u0000<mml:math x","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48063382","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 prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.
{"title":"On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces","authors":"C. Onorati","doi":"10.1090/jag/802","DOIUrl":"https://doi.org/10.1090/jag/802","url":null,"abstract":"In this paper we prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49367741","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 Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q¯ℓoverline {mathbb {Q}}_{ell }-sheaves. This was originally solved by Kerz and Saito in characteristic ≠2neq 2.
{"title":"Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory","authors":"F. Binda, A. Krishna, S. Saito","doi":"10.1090/jag/792","DOIUrl":"https://doi.org/10.1090/jag/792","url":null,"abstract":"<p>We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar Subscript script l\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">overline {mathbb {Q}}_{ell }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-sheaves. This was originally solved by Kerz and Saito in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"not-equals 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>≠<!-- ≠ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">neq 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2020-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44914700","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 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":" ","pages":""},"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}
<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"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </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"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> <p>As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {P}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Hodge-Tate, and we give the first non-trivial numerical checks of the general <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ<!-- χ --></mml:mi> <mml:annotation encoding="application/x-tex">chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathM
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":" ","pages":""},"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":" ","pages":""},"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}
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