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Статті в журналах з теми "Log Fano varieties"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Cheltsov, Ivan A., and Yanir A. Rubinstein. "Asymptotically log Fano varieties." Advances in Mathematics 285 (November 2015): 1241–300. http://dx.doi.org/10.1016/j.aim.2015.08.001.

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2

Fujita, Kento. "Simple normal crossing Fano varieties and log Fano manifolds." Nagoya Mathematical Journal 214 (June 2014): 95–123. http://dx.doi.org/10.1215/00277630-2430136.

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Анотація:

AbstractA projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

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3

Fujita, Kento. "Simple normal crossing Fano varieties and log Fano manifolds." Nagoya Mathematical Journal 214 (June 2014): 95–123. http://dx.doi.org/10.1017/s0027763000010862.

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AbstractA projective log variety (X, D) is called alog Fano manifoldifXis smooth and ifDis a reduced simple normal crossing divisor onΧwith − (KΧ+D) ample. Then-dimensional log Fano manifolds (X, D) with nonzeroDare classified in this article when the log Fano indexrof (X, D) satisfies eitherr≥n/2withρ(X) ≥ 2 orr≥n− 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

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4

Fujita, Kento. "On log K-stability for asymptotically log Fano varieties." Annales de la faculté des sciences de Toulouse Mathématiques 25, no. 5 (2016): 1013–24. http://dx.doi.org/10.5802/afst.1520.

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5

Karzhemanov, I. V. "Semiampleness theorem for weak log Fano varieties." Sbornik: Mathematics 197, no. 10 (October 31, 2006): 1459–65. http://dx.doi.org/10.1070/sm2006v197n10abeh003807.

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6

Schwede, Karl, and Karen E. Smith. "Globally F-regular and log Fano varieties." Advances in Mathematics 224, no. 3 (June 2010): 863–94. http://dx.doi.org/10.1016/j.aim.2009.12.020.

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7

Anderson, Dave, and Alan Stapledon. "Schubert varieties are log Fano over the integers." Proceedings of the American Mathematical Society 142, no. 2 (November 4, 2013): 409–11. http://dx.doi.org/10.1090/s0002-9939-2013-11779-x.

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8

Hassett, Brendan, and Yuri Tschinkel. "Log Fano varieties over function fields of curves." Inventiones mathematicae 173, no. 1 (February 5, 2008): 7–21. http://dx.doi.org/10.1007/s00222-008-0113-2.

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9

Okumura, Katsuhiko. "SNC Log Symplectic Structures on Fano Products." Canadian Mathematical Bulletin 63, no. 4 (February 24, 2020): 891–900. http://dx.doi.org/10.4153/s0008439520000120.

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AbstractThis paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

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10

Lohmann, Daniel. "Families of canonically polarized manifolds over log Fano varieties." Compositio Mathematica 149, no. 6 (March 26, 2013): 1019–40. http://dx.doi.org/10.1112/s0010437x1200053x.

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AbstractLet $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

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11

Berman, Robert J., Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi. "Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 751 (June 1, 2019): 27–89. http://dx.doi.org/10.1515/crelle-2016-0033.

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AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

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12

Namikawa, Yoshinori. "A finiteness theorem on symplectic singularities." Compositio Mathematica 152, no. 6 (April 15, 2016): 1225–36. http://dx.doi.org/10.1112/s0010437x16007387.

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An affine symplectic singularity$X$with a good$\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers$N$and$d$, there are only a finite number of conical symplectic varieties of dimension$2d$with maximal weights$N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

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13

Li, Chi, Xiaowei Wang, and Chenyang Xu. "Algebraicity of the metric tangent cones and equivariant K-stability." Journal of the American Mathematical Society 34, no. 4 (April 9, 2021): 1175–214. http://dx.doi.org/10.1090/jams/974.

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We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

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14

Braun, Lukas. "Gorensteinness and iteration of Cox rings for Fano type varieties." Mathematische Zeitschrift 301, no. 1 (January 9, 2022): 1047–61. http://dx.doi.org/10.1007/s00209-021-02946-w.

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AbstractWe show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones X, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express such X as quotients of a factorial canonical quasicone by a solvable reductive group.

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15

Przyjalkowski, V. V. "On singular log Calabi-Yau compactifications of Landau-Ginzburg models." Sbornik: Mathematics 213, no. 1 (January 1, 2022): 88–108. http://dx.doi.org/10.1070/sm9510.

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Abstract We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index . For coverings of degree greater than the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration, the conjecture that the number of components of the fibre over infinity is equal to the dimension of an anticanonical system of the Fano variety. Bibliography: 46 titles.

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16

Przyjalkowski, V. V. "On singular log Calabi-Yau compactifications of Landau-Ginzburg models." Sbornik: Mathematics 213, no. 1 (January 1, 2022): 88–108. http://dx.doi.org/10.1070/sm9510.

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Анотація:

Abstract We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index . For coverings of degree greater than the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration, the conjecture that the number of components of the fibre over infinity is equal to the dimension of an anticanonical system of the Fano variety. Bibliography: 46 titles.

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17

LAI, CHING-JUI. "BOUNDING VOLUMES OF SINGULAR FANO THREEFOLDS." Nagoya Mathematical Journal 224, no. 1 (October 17, 2016): 37–73. http://dx.doi.org/10.1017/nmj.2016.21.

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Let $(X,\unicode[STIX]{x1D6E5})$ be an $n$-dimensional $\unicode[STIX]{x1D716}$-klt log $\mathbb{Q}$-Fano pair. We give an upper bound for the volume $\text{Vol}(X,\unicode[STIX]{x1D6E5})=(-(K_{X}+\unicode[STIX]{x1D6E5}))^{n}$ when $n=2$, or $n=3$ and $X$ is $\mathbb{Q}$-factorial of $\unicode[STIX]{x1D70C}(X)=1$. This bound is essentially sharp for $n=2$. The main idea is to analyze the covering families of tigers constructed in J. McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov–Alexeev–Borisov Conjecture, which asserts boundedness of the set of $\unicode[STIX]{x1D716}$-klt log $\mathbb{Q}$-Fano varieties of a given dimension $n$.

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18

Zhang, De-Qi. "Polarized endomorphisms of uniruled varieties. With an appendix by Y. Fujimoto and N. Nakayama." Compositio Mathematica 146, no. 1 (December 21, 2009): 145–68. http://dx.doi.org/10.1112/s0010437x09004278.

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AbstractWe show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on ℚ-Fano threefolds, Gorenstein log del Pezzo surfaces and ℙ1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree greater than one is rational.

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19

Nakkajima, Yukiyoshi. "CONGRUENCES OF THE CARDINALITIES OF RATIONAL POINTS OF LOG FANO VARIETIES AND LOG CALABI-YAU VARIETIES OVER THE LOG POINTS OF FINITE FIELDS." Journal of Algebra, Number Theory: Advances and Applications 21, no. 1 (October 10, 2020): 1–51. http://dx.doi.org/10.18642/jantaa_7100122080.

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20

Moraga, Joaquín. "On minimal log discrepancies and kollár components." Proceedings of the Edinburgh Mathematical Society 64, no. 4 (November 2021): 982–1001. http://dx.doi.org/10.1017/s0013091521000729.

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AbstractIn this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities with standard coefficients, which admit an $\epsilon$-plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

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21

Berman, Robert J., and Bo Berndtsson. "Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties." Annales de la faculté des sciences de Toulouse Mathématiques 22, no. 4 (2013): 649–711. http://dx.doi.org/10.5802/afst.1386.

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22

Zhang, Qi. "Rational connectedness of log Q-Fano varieties." Journal fur die reine und angewandte Mathematik (Crelles Journal) 2006, no. 590 (January 26, 2006). http://dx.doi.org/10.1515/crelle.2006.006.

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23

Gongyo, Yoshinori. "On weak Fano varieties with log canonical singularities." Journal für die reine und angewandte Mathematik (Crelles Journal) 2012, no. 665 (January 2012). http://dx.doi.org/10.1515/crelle.2011.111.

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24

Cascini, Paolo, Jesus Martinez-Garcia, and Yanir A. Rubinstein. "On the body of ample angles of asymptotically log Fano varieties." Rendiconti del Circolo Matematico di Palermo Series 2, January 12, 2022. http://dx.doi.org/10.1007/s12215-021-00712-9.

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25

Codogni, Giulio, and Zsolt Patakfalvi. "Positivity of the CM line bundle for families of K-stable klt Fano varieties." Inventiones mathematicae, November 30, 2020. http://dx.doi.org/10.1007/s00222-020-00999-y.

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AbstractThe Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.

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26

Grange, Tim, Elisa Postinghel, and Artie Prendergast-Smith. "Log Fano blowups of mixed products of projective spaces and their effective cones." Revista Matemática Complutense, May 18, 2022. http://dx.doi.org/10.1007/s13163-022-00425-2.

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AbstractWe compute the cones of effective divisors on blowups of $$\mathbb P^1 \times \mathbb P^2$$ P 1 × P 2 and $$\mathbb P^1 \times \mathbb P^3$$ P 1 × P 3 in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for the fact that all the cones we compute are rational polyhedral.

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27

Pukhlikov, Aleksandr V. "Canonical and log canonical thresholds of multiple projective spaces." European Journal of Mathematics, December 2, 2019. http://dx.doi.org/10.1007/s40879-019-00388-7.

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AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

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28

Tseng, Hsian-Hua, and Fenglong You. "A mirror theorem for multi-root stacks and applications." Selecta Mathematica 29, no. 1 (November 2, 2022). http://dx.doi.org/10.1007/s00029-022-00809-8.

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Анотація:

AbstractLet X be a smooth projective variety with a simple normal crossing divisor $$D:=D_1+D_2+\cdots +D_n$$ D : = D 1 + D 2 + ⋯ + D n , where $$D_i\subset X$$ D i ⊂ X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $$X_{D,{\overrightarrow{r}}}$$ X D , r → by constructing an I-function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $$X_{D,\overrightarrow{r}}$$ X D , r → stabilize for sufficiently large $$\overrightarrow{r}$$ r → . (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of $$X_{D,\overrightarrow{r}}$$ X D , r → .

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