Pub Date : 2021-05-04DOI: 10.1215/00127094-2022-0074
M. Mustaţă, S. Olano, M. Popa, J. Witaszek
We study the Du Bois complex $underline{Omega}_Z^bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $widetilde{alpha}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $widetilde{alpha}(Z)geq p+1$, then the canonical morphism $Omega_Z^pto underline{Omega}_Z^p$ is an isomorphism, where $underline{Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $widetilde{alpha}(Z)>pgeq 2$, we obtain non-vanishing results for some of the higher cohomologies of $underline{Omega}_Z^{n-p}$.
{"title":"The Du Bois complex of a hypersurface and the minimal exponent","authors":"M. Mustaţă, S. Olano, M. Popa, J. Witaszek","doi":"10.1215/00127094-2022-0074","DOIUrl":"https://doi.org/10.1215/00127094-2022-0074","url":null,"abstract":"We study the Du Bois complex $underline{Omega}_Z^bullet$ of a hypersurface $Z$ in a smooth complex algebraic variety in terms its minimal exponent $widetilde{alpha}(Z)$. The latter is an invariant of singularities, defined as the negative of the greatest root of the reduced Bernstein-Sato polynomial of $Z$, and refining the log canonical threshold. We show that if $widetilde{alpha}(Z)geq p+1$, then the canonical morphism $Omega_Z^pto underline{Omega}_Z^p$ is an isomorphism, where $underline{Omega}_Z^p$ is the $p$-th associated graded piece of the Du Bois complex with respect to the Hodge filtration. On the other hand, if $Z$ is singular and $widetilde{alpha}(Z)>pgeq 2$, we obtain non-vanishing results for some of the higher cohomologies of $underline{Omega}_Z^{n-p}$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46285535","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}
Pub Date : 2021-03-30DOI: 10.1215/00127094-2022-0079
B. Yu
The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2. if r is not an integer, M(r) does not carry any Anosov flow. As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds which carry taut foliations but does not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on M(r).
{"title":"Anosov flows on Dehn surgeries on the figure-eight knot","authors":"B. Yu","doi":"10.1215/00127094-2022-0079","DOIUrl":"https://doi.org/10.1215/00127094-2022-0079","url":null,"abstract":"The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2. if r is not an integer, M(r) does not carry any Anosov flow. As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds which carry taut foliations but does not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on M(r).","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42019788","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}
Pub Date : 2021-03-16DOI: 10.1215/00127094-2022-0028
Adrien Boulanger, G. Courtois
We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.
我们给出并证明了闭可定向黎曼流形上Coexat1-形式的一个类Cheeger不等式。
{"title":"A Cheeger-like inequality for coexact 1-forms","authors":"Adrien Boulanger, G. Courtois","doi":"10.1215/00127094-2022-0028","DOIUrl":"https://doi.org/10.1215/00127094-2022-0028","url":null,"abstract":"We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45100321","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}
Pub Date : 2021-03-02DOI: 10.1215/00127094-2022-0068
P. Etingof, E. Frenkel, D. Kazhdan
We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n).
{"title":"Hecke operators and analytic Langlands correspondence for curves over local fields","authors":"P. Etingof, E. Frenkel, D. Kazhdan","doi":"10.1215/00127094-2022-0068","DOIUrl":"https://doi.org/10.1215/00127094-2022-0068","url":null,"abstract":"We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n).","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46878067","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}
Pub Date : 2021-02-21DOI: 10.1215/00127094-2022-0086
A. Cohen, Guy Moshkovitz
We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to 1+o(1) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary we prove, allowing the field to depend on the value of the norm, the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.
{"title":"Partition and analytic rank are equivalent over large fields","authors":"A. Cohen, Guy Moshkovitz","doi":"10.1215/00127094-2022-0086","DOIUrl":"https://doi.org/10.1215/00127094-2022-0086","url":null,"abstract":"We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to 1+o(1) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary we prove, allowing the field to depend on the value of the norm, the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44567506","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}
Pub Date : 2021-02-19DOI: 10.1215/00127094-2022-0031
Jean-François Babadjian, F. Iurlano, F. Rindler
This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size $varepsilon downarrow 0$ to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitt'e in 2001 and predicted heuristically before in works of Allaire&Kohn and Kohn&Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, settling this open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) $mathrm{div}$-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang integrand, and the characterization of limit minimizers due to Bouchitt'e&Buttazzo.
{"title":"Shape optimization of light structures and the vanishing mass conjecture","authors":"Jean-François Babadjian, F. Iurlano, F. Rindler","doi":"10.1215/00127094-2022-0031","DOIUrl":"https://doi.org/10.1215/00127094-2022-0031","url":null,"abstract":"This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size $varepsilon downarrow 0$ to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitt'e in 2001 and predicted heuristically before in works of Allaire&Kohn and Kohn&Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, settling this open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) $mathrm{div}$-quasiconvex quadratic forms, computations involving the Hashin-Shtrikman bounds for the Kohn-Strang integrand, and the characterization of limit minimizers due to Bouchitt'e&Buttazzo.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49220942","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}
Pub Date : 2021-02-15DOI: 10.1215/00127094-2020-0048
G. David, Max Engelstein, S. Mayboroda
We characterize the rectifiability (both uniform and not) of an Ahlfors regular set E of arbitrary codimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in sharp contrast with the usual setting of codimension 1, a converse to Dahlberg’s theorem must be false on lower-dimensional boundaries without additional assumptions.
{"title":"Square functions, nontangential limits, and harmonic measure in codimension larger than 1","authors":"G. David, Max Engelstein, S. Mayboroda","doi":"10.1215/00127094-2020-0048","DOIUrl":"https://doi.org/10.1215/00127094-2020-0048","url":null,"abstract":"We characterize the rectifiability (both uniform and not) of an Ahlfors regular set E of arbitrary codimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that, in sharp contrast with the usual setting of codimension 1, a converse to Dahlberg’s theorem must be false on lower-dimensional boundaries without additional assumptions.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":"170 1","pages":"455-501"},"PeriodicalIF":2.5,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41499231","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}
Pub Date : 2021-02-08DOI: 10.1215/00127094-2022-0035
Espen Bernton, Promit Ghosal, Marcel Nutz
We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.
{"title":"Entropic optimal transport: Geometry and large deviations","authors":"Espen Bernton, Promit Ghosal, Marcel Nutz","doi":"10.1215/00127094-2022-0035","DOIUrl":"https://doi.org/10.1215/00127094-2022-0035","url":null,"abstract":"We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42394567","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}
Pub Date : 2021-01-28DOI: 10.1215/00127094-2022-0058
V. Blomer, S. Jana, Paul D. Nelson
Let $pi_1, pi_2, pi_3$ be three cuspidal automorphic representations for the group ${rm SL}(2, Bbb{Z})$, where $pi_1$ and $pi_2$ are fixed and $pi_3$ has large conductor. We prove a subconvex bound for $L(1/2, pi_1 otimes pi_2 otimes pi_3)$ of Weyl-type quality. Allowing $pi_3$ to be an Eisenstein series we also obtain a Weyl-type subconvex bound for $L(1/2 + it, pi_1 otimes pi_2)$.
{"title":"The Weyl bound for triple product L-functions","authors":"V. Blomer, S. Jana, Paul D. Nelson","doi":"10.1215/00127094-2022-0058","DOIUrl":"https://doi.org/10.1215/00127094-2022-0058","url":null,"abstract":"Let $pi_1, pi_2, pi_3$ be three cuspidal automorphic representations for the group ${rm SL}(2, Bbb{Z})$, where $pi_1$ and $pi_2$ are fixed and $pi_3$ has large conductor. We prove a subconvex bound for $L(1/2, pi_1 otimes pi_2 otimes pi_3)$ of Weyl-type quality. Allowing $pi_3$ to be an Eisenstein series we also obtain a Weyl-type subconvex bound for $L(1/2 + it, pi_1 otimes pi_2)$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42461712","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}
Pub Date : 2021-01-22DOI: 10.1215/00127094-2022-0026
Hamid Abban, Ziquan Zhuang
We give a lower bound of the $delta$-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.
{"title":"Seshadri constants and K-stability of Fano manifolds","authors":"Hamid Abban, Ziquan Zhuang","doi":"10.1215/00127094-2022-0026","DOIUrl":"https://doi.org/10.1215/00127094-2022-0026","url":null,"abstract":"We give a lower bound of the $delta$-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47470137","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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