Pub Date : 2021-08-17DOI: 10.4007/annals.2023.198.2.7
Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro, Z. Patakfalvi
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.
在适当的条件下，我们证明了当因子的系数变化时，在所有维度上稳定对数对的模空间都存在穿墙和归约态射。
{"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}
Pub Date : 2021-07-30Print Date: 2021-10-01DOI: 10.3171/2020.11.SPINE201313
Heiko Koller, Meric Enercan, Sebastian Decker, Hossein Mehdian, Luigi Aurelio Nasto, Wolfgang Hitzl, Juliane Koller, Axel Hempfing, Azmi Hamzaoglu
Objective: 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.
Methods: 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).
Results: 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%).
Conclusions: 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.
{"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}
Pub Date : 2021-07-12DOI: 10.4007/annals.2023.198.2.6
Will Hide, Michael Magee
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 $frac{1}{4}$.
证明了如果$X$是一个有限面积的非紧双曲曲面，那么对于任意$epsilon>0$，当概率趋近于1为$ntoinfty$时，$X$的一致随机度$n$黎曼覆盖除了$X$的特征值外，没有$[0,frac{1}{4}-epsilon)$的拉普拉斯特征值，并且具有相同的多重性。结果，利用Buser, Burger, and Dodziuk的紧化过程，我们肯定地解决了是否存在一类闭双曲曲面序列的问题，这些曲面的属趋于无穷，且拉普拉斯算子的第一非零特征值趋于$frac{1}{4}$。
{"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}
Pub Date : 2021-07-06DOI: 10.4007/annals.2023.197.3.5
K. Coulembier, P. Etingof, V. Ostrik
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.
P. Deligne(2002)的一个基本定理指出，特征为零的代数闭域上的前tannakian范畴允许光纤函子进入超向量空间的范畴(即，是仿射原代数超群的表示范畴)，当且仅当它具有适度增长(即，一个对象的张量幂的长度最多以指数增长)。本文证明了该定理的一个特征p版本。也就是说，我们证明了特征为p>0的代数闭域上的一个前tannakian范畴允许一个纤维函子进入Verlinde范畴Ver_p(即，是Ver_p中仿射群方案的表示范畴)，当且仅当它具有适度增长并且是Frobenius精确的。这意味着Frobenius精确的前tannakian适度增长范畴承认Frobenius- perron维度的良好表现。由此可见，任何中等增长的半简单前tannakian范畴都有一个到Ver_p的纤维函子(因此Deligne定理特别适用于特征2,3的半简单前tannakian范畴)。这就解决了2015年第三位作者的猜想。特别地，这个结果适用于有限群的模表示(或更一般地，仿射群格式)的范畴的半简化，这给经典模表示理论提供了新的应用。例如，它允许我们描述，对于一个模表示V，在V^{o * n}中，维数从素数到p的不可分解和的数目的可能增长率。
{"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}
Pub Date : 2021-06-04DOI: 10.4007/annals.2023.198.1.4
B. Guo, D. Phong, Freid Tong
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.
{"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}
Pub Date : 2021-02-23DOI: 10.4007/annals.2021.194.3.9
Giles Gardam
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.
{"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}
Pub Date : 2021-02-18DOI: 10.4007/annals.2022.196.2.2
Yuchen Liu, Chenyang Xu, Ziquan Zhuang
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.
{"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}
Pub Date : 2021-02-12DOI: 10.4007/annals.2023.198.2.4
Paul Apisa, A. Wright
We classify GL(2,R) orbit closures of translation surfaces of rank at least half the genus plus 1.
我们对秩至少为半属加1的平移曲面的GL(2,R)轨道闭包进行了分类。
{"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}
Pub Date : 2021-02-02DOI: 10.4007/annals.2022.195.3.5
D. Ryabogin
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 $dge 3$.
{"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}
Pub Date : 2021-01-28DOI: 10.4007/annals.2023.198.2.5
V. Alexeev, P. Engel
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. For 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 $d$.
{"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}
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：