Positivity of the CM line bundle for families of K-stable klt Fano varieties

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.

On the Central Limit Theorem of Probability Theory. II

Selected Papers I ◽

10.1007/978-3-319-16859-3_17 ◽

2015 ◽

pp. 403-416

Author(s):

William Feller

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

The Patterns of Large Numbers

The Error of Truth ◽

10.1093/oso/9780198831600.003.0004 ◽

2019 ◽

pp. 43-66

Author(s):

Steven J. Osterlind

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

French Revolution ◽

Central Limit ◽

Historical Context ◽

Binomial Theorem ◽

Large Numbers ◽

First Time

This chapter advances the historical context for quantification by describing the climate of the day—social, cultural, political, and intellectual—as fraught with disquieting influences. Forces leading to the French Revolution were building, and the colonists in America were fighting for secession from England. During this time, three important number theorems came into existence: the binomial theorem, the law of large numbers, and the central limit theorem. Each is described in easy-to-understand language. These are fundamental to how numbers operate in a probability circumstance. Pascal’s triangle is explained as a shortcut solving some binomial expansions, and Jacob Bernoulli’s Ars Conjectandi, which presents the study of measurement “error” for the first time, is discussed. In addition, the central limit theorem is explained in terms of its relevance to probability theory, and its utility today.

Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

Journal of the American Mathematical Society ◽

10.1090/s0894-0347-06-00528-5 ◽

2006 ◽

Vol 19 (3) ◽

pp. 579-623 ◽

Cited By ~ 24

Author(s):

Alexander I. Bufetov

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Moduli Space ◽

Decay Of Correlations ◽

Interval Exchange ◽

Abelian Differentials ◽

Interval Exchange Transformations ◽

Teichmüller Flow

A principle of subsequences in probability theory: The central limit theorem

Advances in Mathematics ◽

10.1016/0001-8708(74)90064-4 ◽

1974 ◽

Vol 13 (1) ◽

pp. 31-54 ◽

Cited By ~ 18

Author(s):

S.D Chatterji

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

The Kagan Classification of Multivariate Distributions and the Central Limit Theorem

Recent Advances in Statistics and Probability ◽

10.1515/9783112313961-008 ◽

1994 ◽

pp. 67-70

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Multivariate Distributions ◽

Probability for Everyone—Even Philosophers

10.1093/oxfordhb/9780199607617.013.2 ◽

2017 ◽

Author(s):

Alan Hájek ◽

Christopher Hitchcock

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

Conditional Probability ◽

Central Limit ◽

Measure Theory ◽

Bayes Theorem ◽

Large Numbers ◽

Key Concepts

In this chapter the basics of probability theory are introduced, with particular attention to those topics that are most important for applications in philosophy. The formalism is described in two passes. The first presents finite probability, which suffices for most philosophical discussions of probability. The second presents measure theory, which is needed for applications involving infinities or limits. Key concepts such as conditional probability, probabilistic independence, random variables, and expectation are defined. In addition, several important theorems, including Bayes’ theorem, the weak and strong laws of large numbers, and the central limit theorem are defined. Along the way, several familiar puzzles or paradoxes involving probability are discussed.

On the Central Limit Theorem of Probability Theory

Selected Papers I ◽

10.1007/978-3-319-16859-3_12 ◽

2015 ◽

pp. 207-244

Author(s):

William Feller

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

On non-additive probability measures

Mathematica Slovaca ◽

10.1515/ms-2017-0071 ◽

2017 ◽

Vol 67 (6) ◽

Author(s):

Beloslav Riečan ◽

Karol Samuelčik

Keyword(s):

Probability Theory ◽

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Mathematical Analysis ◽

Independent Random Variables ◽

Probability Measures ◽

Additive Probability ◽

Gauss Distribution

AbstractThe additivity of considered measures or integrals resp. can be omitted in some problems of mathematical analysis and its applications. In the paper it is shown that similar situations are possible also in the probability theory. As an example is proved a generalized version of the central limit theorem about the convergence of arithmetical means of independent random variables to the Gauss distribution.