Annals of Mathematics最新文献

{"title":"Wall crossing for moduli of stable log pairs","authors":"Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro, Z. Patakfalvi","doi":"10.4007/annals.2023.198.2.7","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.7","url":null,"abstract":"We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45811990","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":"Resolution of the lumbosacral fractional curve and evaluation of the risk for adding on in 101 patients with posterior correction of Lenke 3, 4, and 6 curves.","authors":"Heiko Koller, Meric Enercan, Sebastian Decker, Hossein Mehdian, Luigi Aurelio Nasto, Wolfgang Hitzl, Juliane Koller, Axel Hempfing, Azmi Hamzaoglu","doi":"10.3171/2020.11.SPINE201313","DOIUrl":"10.3171/2020.11.SPINE201313","url":null,"abstract":"<p><strong>Objective: </strong>In double and triple major adolescent idiopathic scoliosis curves it is still controversial whether the lowest instrumented vertebra (LIV) should be L3 or L4. Too short a fusion can impede postoperative distal curve compensation and promote adding on (AON). Longer fusions lower the chance of compensation by alignment changes of the lumbosacral curve (LSC). This study sought to improve prediction accuracy for AON and surgical outcomes in Lenke type 3, 4, and 6 curves.</p><p><strong>Methods: </strong>This was a retrospective multicenter analysis of patients with adolescent idiopathic scoliosis who had Lenke 3, 4, and 6 curves and ≥ 1 year of follow-up after posterior correction. Resolution of the LSC was studied by changes of LIV tilt, L3 tilt, and L4 tilt, with the variables resembling surrogate measures for the LSC. AON was defined as a disc angle below LIV > 5° at follow-up. A matched-pairs analysis was done of differences between LIV at L3 and at L4. A multivariate prediction analysis evaluated the AON risk in patients with LIV at L3. Clinical outcomes were assessed by the Scoliosis Research Society 22-item questionnaire (SRS-22).</p><p><strong>Results: </strong>The sample comprised 101 patients (average age 16 years). The LIV was L3 in 54%, and it was L4 in 39%. At follow-up, 87% of patients showed shoulder balance, 86% had trunk balance, and 64% had a lumbar curve (LC) ≤ 20°. With an LC ≤ 20° (p = 0.01), SRS-22 scores were better and AON was less common (26% vs 59%, p = 0.001). Distal extension of the fusion (e.g., LIV at L4) did not have a significant influence on achieving an LSC < 20°; however, higher screw density allowed better LC correction and resulted in better spontaneous LSC correction. AON occurred in 34% of patients, or 40% if the LIV was L3. Patients with AON had a larger residual LSC, worse LC correction, and worse thoracic curve (TC) correction. A total of 44 patients could be included in the matched-pairs analysis. LC correction and TC correction were comparable, but AON was 50% for LIV at L3 and 18% for LIV at L4. Patients without AON had a significantly better LC correction and TC correction (p < 0.01). For patients with LIV at L3, a significant prediction model for AON was established including variables addressed by surgeons: postoperative LC and TC (negative predictive value 78%, positive predictive value 79%, sensitivity 79%, specificity 81%).</p><p><strong>Conclusions: </strong>An analysis of 101 patients with Lenke 3, 4, and 6 curves showed that TC and LC correction had significant influence on LSC resolution and the risk for AON. Improving LC correction and achieving an LC < 20° offers the potential to lower the risk for AON, particularly in patients with LIV at L3.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"85 1","pages":"471-485"},"PeriodicalIF":2.9,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90531363","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":"Near optimal spectral gaps for hyperbolic surfaces","authors":"Will Hide, Michael Magee","doi":"10.4007/annals.2023.198.2.6","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.6","url":null,"abstract":"We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $epsilon>0$, with probability tending to one as $ntoinfty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,frac{1}{4}-epsilon)$ other than those of $X$, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $frac{1}{4}$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42326979","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":"On Frobenius exact symmetric tensor categories","authors":"K. Coulembier, P. Etingof, V. Ostrik","doi":"10.4007/annals.2023.197.3.5","DOIUrl":"https://doi.org/10.4007/annals.2023.197.3.5","url":null,"abstract":"A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde category Ver_p (i.e., is the representation category of an affine group scheme in Ver_p) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well-behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderate growth has a fiber functor to Ver_p (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics 2,3). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation V, the possible growth rates of the number of indecomposable summands in V^{otimes n} of dimension prime to p.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46722499","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":"On $L^infty$ estimates for complex Monge-Ampère equations","authors":"B. Guo, D. Phong, Freid Tong","doi":"10.4007/annals.2023.198.1.4","DOIUrl":"https://doi.org/10.4007/annals.2023.198.1.4","url":null,"abstract":"A PDE proof is provided for the sharp $L^infty$ estimates for the complex Monge-Amp`ere equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42008015","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":"A counterexample to the unit conjecture for group rings","authors":"Giles Gardam","doi":"10.4007/annals.2021.194.3.9","DOIUrl":"https://doi.org/10.4007/annals.2021.194.3.9","url":null,"abstract":"The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43370264","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":"Finite generation for valuations computing stability thresholds and applications to K-stability","authors":"Yuchen Liu, Chenyang Xu, Ziquan Zhuang","doi":"10.4007/annals.2022.196.2.2","DOIUrl":"https://doi.org/10.4007/annals.2022.196.2.2","url":null,"abstract":"We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies: (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of K\"ahler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41954772","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":"High rank invariant subvarieties","authors":"Paul Apisa, A. Wright","doi":"10.4007/annals.2023.198.2.4","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.4","url":null,"abstract":"We classify GL(2,R) orbit closures of translation surfaces of rank at least half the genus plus 1.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46904418","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":"A negative answer to Ulam's Problem 19 from the Scottish Book","authors":"D. Ryabogin","doi":"10.4007/annals.2022.195.3.5","DOIUrl":"https://doi.org/10.4007/annals.2022.195.3.5","url":null,"abstract":"We give a negative answer to Ulam's Problem 19 from the Scottish Book asking {it is a solid of uniform density which will float in water in every position a sphere?} Assuming that the density of water is $1$, we show that there exists a strictly convex body of revolution $Ksubset {mathbb R^3}$ of uniform density $frac{1}{2}$, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions $dge 3$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43787054","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":"Compact moduli of K3 surfaces","authors":"V. Alexeev, P. Engel","doi":"10.4007/annals.2023.198.2.5","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.5","url":null,"abstract":"Let $F$ be a moduli space of lattice-polarized K3 surfaces. Suppose that one has chosen a canonical effective ample divisor $R$ on a general K3 in $F$. We call this divisor \"recognizable\" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification $overline{F}^R$ for a recognizable divisor is a Looijenga semitoroidal compactification. \u0000For polarized K3 surfaces $(X,L)$ of degree $2d$, we show that the sum of rational curves in the linear system $|L|$ is a recognizable divisor, giving a modular semitoroidal compactification of $F_{2d}$ for all $d$.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44100100","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}