On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a C ∗ mathbb {C}^* action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.
{"title":"Vafa-Witten invariants for projective surfaces I: stable case","authors":"Yuuji Tanaka, Richard P. Thomas","doi":"10.1090/JAG/738","DOIUrl":"https://doi.org/10.1090/JAG/738","url":null,"abstract":"On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a \u0000\u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 ∗\u0000 \u0000 mathbb {C}^*\u0000 \u0000\u0000 action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations.\u0000\u0000When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/738","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49629199","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 an Atiyah-Segal isomorphism for the higher K K -theory of coherent sheaves on quotient Deligne-Mumford stacks over C mathbb {C} . As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher K K -theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.
{"title":"Atiyah-Segal theorem for Deligne-Mumford stacks and applications","authors":"A. Krishna, Bhamidi Sreedhar","doi":"10.1090/jag/755","DOIUrl":"https://doi.org/10.1090/jag/755","url":null,"abstract":"We prove an Atiyah-Segal isomorphism for the higher \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theory of coherent sheaves on quotient Deligne-Mumford stacks over \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 mathbb {C}\u0000 \u0000\u0000. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/755","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46023833","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 toric b b -divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric b b -divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b b -divisor is equal to the number of lattice points in this convex set and we give a Hilbert–Samuel-type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to b b -divisors with Newton–Okounkov bodies. The main motivation for studying toric b b -divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.
{"title":"Intersection theory of toric 𝑏-divisors in toric varieties","authors":"A. M. Botero","doi":"10.1090/JAG/721","DOIUrl":"https://doi.org/10.1090/JAG/721","url":null,"abstract":"We introduce toric \u0000\u0000 \u0000 b\u0000 b\u0000 \u0000\u0000-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric \u0000\u0000 \u0000 b\u0000 b\u0000 \u0000\u0000-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric \u0000\u0000 \u0000 b\u0000 b\u0000 \u0000\u0000-divisor is equal to the number of lattice points in this convex set and we give a Hilbert–Samuel-type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to \u0000\u0000 \u0000 b\u0000 b\u0000 \u0000\u0000-divisors with Newton–Okounkov bodies. The main motivation for studying toric \u0000\u0000 \u0000 b\u0000 b\u0000 \u0000\u0000-divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/721","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45799552","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":"The class of the affine line is a zero divisor in the Grothendieck ring","authors":"L. Borisov","doi":"10.7282/T33B62H9","DOIUrl":"https://doi.org/10.7282/T33B62H9","url":null,"abstract":"","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78792507","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}
For every fibration f:X→Bf : X to B with XX a compact Kähler manifold, BB a smooth projective curve, and a general fiber of ff an abelian variety, we prove that ff has an algebraic approximation.
对于每一个纤维f: X→B f: X 到B，其中X X是紧的Kähler流形，B B是光滑的投影曲线，f f是一个一般的纤维和阿贝尔变，我们证明了f f有一个代数逼近。
{"title":"Algebraic approximations of fibrations in abelian varieties over a curve","authors":"Hsueh-Yung Lin","doi":"10.1090/jag/791","DOIUrl":"https://doi.org/10.1090/jag/791","url":null,"abstract":"<p>For every fibration <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper B\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>B</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f : X to B</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with <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> a compact Kähler manifold, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\">\u0000 <mml:semantics>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">B</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> a smooth projective curve, and a general fiber of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> an abelian variety, we prove that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has an algebraic approximation.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551290","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 an algebraic version of classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove that the number of fixed points cannot be odd. The main tool is a construction originally used by Rost in the context of the degree formula. The framework of diagonalisable groups allows us to include the case of base fields of characteristic p.
{"title":"Diagonalisable $p$-groups cannot fix exactly one point on projective varieties","authors":"Olivier Haution","doi":"10.1090/jag/749","DOIUrl":"https://doi.org/10.1090/jag/749","url":null,"abstract":"We prove an algebraic version of classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove that the number of fixed points cannot be odd. The main tool is a construction originally used by Rost in the context of the degree formula. The framework of diagonalisable groups allows us to include the case of base fields of characteristic p.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/749","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550964","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 if f : X → Z is a smooth surjective morphism between projective manifolds and if −KX is semi-ample, then −KZ is also semi-ample. This was conjectured by Fujino and Gongyo. We list several counter-examples to show that this fails without the smoothness assumption on f . We prove the above result by proving some results concerning the moduli divisor of the canonical bundle formula associated to a klt-trivial fibration (X,B)→ Z.
{"title":"Images of manifolds with semi-ample anti-canonical divisor","authors":"C. Birkar, Yifei Chen","doi":"10.1090/JAG/662","DOIUrl":"https://doi.org/10.1090/JAG/662","url":null,"abstract":"We prove that if f : X → Z is a smooth surjective morphism between projective manifolds and if −KX is semi-ample, then −KZ is also semi-ample. This was conjectured by Fujino and Gongyo. We list several counter-examples to show that this fails without the smoothness assumption on f . We prove the above result by proving some results concerning the moduli divisor of the canonical bundle formula associated to a klt-trivial fibration (X,B)→ Z.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/662","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550551","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 RR be a Cohen–Macaulay normal domain with a canonical module ωRomega _R. It is proved that if RR admits a noncommutative crepant resolution (NCCR), then necessarily it is Qmathds {Q}-Gorenstein. Writing SS for a Zariski local canonical cover of RR, a tight relationship between the existence of noncommutative (crepant) resolutions on RR and SS is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-termin
{"title":"Gorenstein modifications and mathds{𝑄}-Gorenstein rings","authors":"Hailong Dao, O. Iyama, Ryo Takahashi, M. Wemyss","doi":"10.1090/JAG/760","DOIUrl":"https://doi.org/10.1090/JAG/760","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 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> be a Cohen–Macaulay normal domain with a canonical module <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"omega Subscript upper R\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>ω<!-- ω --></mml:mi>\u0000 <mml:mi>R</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">omega _R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. It is proved that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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> admits a noncommutative crepant resolution (NCCR), then necessarily it is <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\">mathds {Q}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Gorenstein. Writing <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> for a Zariski local canonical cover of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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>, a tight relationship between the existence of noncommutative (crepant) resolutions on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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> and <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> is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-termin","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/760","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551255","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 study families of algebraic spaces with G m {mathbb G}_m -action and prove Braden’s theorem on hyperbolic localization for arbitrary base schemes. As an application, we obtain that hyperbolic localization commutes with nearby cycles.
研究了具有G m {mathbb G}_m -作用的代数空间族，并证明了任意基格式双曲局部化的Braden定理。作为一个应用，我们得到了双曲局部化与附近环的交换。
{"title":"Spaces with 𝔾_{𝕞}-action, hyperbolic localization and nearby cycles","authors":"Timo Richarz","doi":"10.1090/jag/710","DOIUrl":"https://doi.org/10.1090/jag/710","url":null,"abstract":"We study families of algebraic spaces with \u0000\u0000 \u0000 \u0000 \u0000 \u0000 G\u0000 \u0000 \u0000 m\u0000 \u0000 {mathbb G}_m\u0000 \u0000\u0000-action and prove Braden’s theorem on hyperbolic localization for arbitrary base schemes. As an application, we obtain that hyperbolic localization commutes with nearby cycles.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/710","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550443","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}
The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets.
{"title":"Algebraic stratified general position and transversality","authors":"C. McCrory, A. Parusiński, L. Paunescu","doi":"10.1090/JAG/713","DOIUrl":"https://doi.org/10.1090/JAG/713","url":null,"abstract":"The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/713","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550938","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：