We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
{"title":"Parallel transport for vector bundles on 𝑝-adic varieties","authors":"C. Deninger, A. Werner","doi":"10.1090/jag/747","DOIUrl":"https://doi.org/10.1090/jag/747","url":null,"abstract":"We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/747","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49655222","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 X be a simply connected projective manifold with nef anticanonical bundle. We prove that X X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC-fibration of any projective manifold with nef anticanonical bundle.
{"title":"A decomposition theorem for projective manifolds with nef anticanonical bundle","authors":"Junyan Cao, A. Horing","doi":"10.1090/JAG/715","DOIUrl":"https://doi.org/10.1090/JAG/715","url":null,"abstract":"Let \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 be a simply connected projective manifold with nef anticanonical bundle. We prove that \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC-fibration of any projective manifold with nef anticanonical bundle.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/715","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44568805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2). The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).
{"title":"Boundedness questions for Calabi–Yau threefolds","authors":"P. Wilson","doi":"10.1090/JAG/781","DOIUrl":"https://doi.org/10.1090/JAG/781","url":null,"abstract":"In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms.\u0000\u0000Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2).\u0000\u0000The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44857653","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":"PD Higgs crystals and Higgs cohomology in characteristic","authors":"Hideto Oyama","doi":"10.1090/JAG/699","DOIUrl":"https://doi.org/10.1090/JAG/699","url":null,"abstract":"","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"26 1","pages":"735-802"},"PeriodicalIF":1.8,"publicationDate":"2017-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/699","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47044659","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 cone of primitive Heegner divisors is finitely generated for many orthogonal Shimura varieties, including the moduli space of polarized K 3 K3 -surfaces. The proof relies on the growth of coefficients of modular forms.
{"title":"Cones of Heegner divisors","authors":"J. Bruinier, M. Moller","doi":"10.1090/JAG/734","DOIUrl":"https://doi.org/10.1090/JAG/734","url":null,"abstract":"We show that the cone of primitive Heegner divisors is finitely generated for many orthogonal Shimura varieties, including the moduli space of polarized \u0000\u0000 \u0000 \u0000 K\u0000 3\u0000 \u0000 K3\u0000 \u0000\u0000-surfaces. The proof relies on the growth of coefficients of modular forms.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/734","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48079677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that p p -adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.
{"title":"On rigid varieties with projective reduction","authors":"Shizhang Li","doi":"10.1090/jag/740","DOIUrl":"https://doi.org/10.1090/jag/740","url":null,"abstract":"In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/740","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49053961","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 associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type ( d , … , d ) (d,dots , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.
{"title":"Stability of associated forms","authors":"M. Fedorchuk, A. Isaev","doi":"10.1090/JAG/719","DOIUrl":"https://doi.org/10.1090/JAG/719","url":null,"abstract":"We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type \u0000\u0000 \u0000 \u0000 (\u0000 d\u0000 ,\u0000 …\u0000 ,\u0000 d\u0000 )\u0000 \u0000 (d,dots , d)\u0000 \u0000\u0000 is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/719","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49202850","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}
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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
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