![]() ![]() Lawrence Ein, Robert Lazarsfeld, and Karen E.Étude cohomologique des faisceaux cohérents. ![]() Grothendieck, Éléments de géométrie algébrique. Simon Donaldson and Song Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math.Tommaso de Fernex, Lawrence Ein, and Mircea Mustaţă, Log canonical thresholds on varieties with bounded singularities, Classification of algebraic varieties, EMS Ser.Tommaso de Fernex, Lawrence Ein, and Mircea Mustaţă, Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties, Duke Math.III: Limits as cone angle approaches $2\pi$ and completion of the main proof, J. Xiuxiong Chen, Simon Donaldson, and Song Sun, Kähler-Einstein metrics on Fano manifolds.Weichung Chen, Gabriele Di Cerbo, Jingjun Han, Chen Jiang, and Roberto Svaldi, Birational boundedness of rationally connected Calabi-Yau 3-folds, Adv.Guodu Chen and Jingjun Han, Boundedness of $(\epsilon, n)$-complements for surfaces, Adv.Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.Lukas Braun, The local fundamental group of a Kawamata log terminal singularity is finite, Invent.Sébastien Boucksom, Charles Favre, and Mattias Jonsson, A refinement of Izumi’s theorem, Valuation theory in interaction, EMS Ser.Lecture Note Ser., vol. 417, Cambridge Univ. Urbinati, Valuation spaces and multiplier ideals on singular varieties, Recent advances in algebraic geometry, London Math. Xu, Openness of K-semistability for Fano varieties, Duke Math. Harold Blum and Yuchen Liu, The normalized volume of a singularity is lower semicontinuous, J.Liu, Openness of uniform K-stability in families of $\mathbb $-Fano varieties, Ann. Harold Blum and Mattias Jonsson, Thresholds, valuations, and K-stability, Adv.Blum, Singularities and K-stability, Ph.D. Harold Blum, Existence of valuations with smallest normalized volume, Compos.Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Caucher Birkar, Paolo Cascini, Christopher D.Caucher Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann.Caucher Birkar, Anti-pluricanonical systems on Fano varieties, Ann.Berman, Sébastien Boucksom, and Mattias Jonsson, A variational approach to the Yau-Tian-Donaldson conjecture, J. Berman, Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. Liu, K-moduli of curves on a quadric surface and K3 surfaces, J. Liu, Wall crossing for K-moduli spaces of plane curves, arXiv: 1909.04576, 2019. Artin, Algebraic approximation of structures over complete local rings, Inst. Florin Ambro, Variation of log canonical thresholds in linear systems, Int.Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities. We show that the latter conjecture also holds when the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. ACC for local volumes and boundedness of singularitiesĪuthors: Jingjun Han, Yuchen Liu and Lu QiĪbstract: The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities $x\in (X,\Delta )$ satisfies the ACC if the coefficients of $\Delta$ belong to a descending chain condition (DCC) set. ![]()
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