The dual complex of Calabi–Yau pairs

By J. Kollár. Inventiones Mathematicae, 2015

89 citations recorded by the source corpus.

A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that KX+D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_X+D$$\end{document} is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of D is a quotient of the fundamental group of the smooth locus of X, hence its pro-finite completion is finite. This leads to a positive answer in dimension ≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le $$\end{document}4. We also study the dual complex of degenerations of Calabi–Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of D supports an ample divisor.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a3d53952d773fd2a4151492d4a7eff806b6f9768

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Tropical and non-Archimedean limits of degenerating families of volume forms (2016, background): “aximally degenerate case d= n, if Xhas semistable reduction, then Sk(X) is even a pseudomanifold, i.e. (a) is replaced by (a’) every (n 1)-simplex of is contained in exactly two n-simplices. See also [KX15] for even more precise results on the structure of Sk(X). For example, Sk(X) is homeomorphic t”
Theta functions on varieties with effective anti-canonical class (2016, background): “typical situation in which such a Barises is as the dual intersection complex of THETA FUNCTIONS 5 a dlt minimal model of a maximally unipotent degeneration of Calabi-Yau varieties. Koll´ar and Xu in [KoXu], §33 showed that Bwill indeed be a manifold oﬀ of a codimension three subset (see also [NX] f”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, methodology): “[42, 59, 64]) treat the general K-trivial case.”