Fibers over infinity of Landau–Ginzburg models

We conjecture that the number of components of the fiber over infinity of Landau–Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for $\log$ Calabi–Yau compactifications of toric Landau–Ginzburg models for smooth Fano threefolds, complete intersections in projective spaces, and some toric varieties.

Keywords

Fano varieties, Landau–Ginzburg models, $\log$ Calabi–Yau compactifications, anticanonical linear systems

2010 Mathematics Subject Classification

Primary 14J45. Secondary 14J33.

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Ivan Cheltsov was supported by the EPSRC Grant Number EP/V054597/1.

The work of V. V. Przyjalkowski was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265). He is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.

Received 21 December 2021

Accepted 18 July 2022

Published 21 October 2022