publications

2026

1. Preprint

   Boundedness of polarized foliated surfaces

   Yen-An Chen, and Minzhe Zhu

   Jul 2026

   arXiv:2607.28586

   Abs arXiv

   We establish the minimal model program for klt adjoint foliated surfaces and use it to study the boundedness of polarized adjoint foliated surfaces. We prove that ε-lc adjoint foliated surfaces with nef adjoint canonical divisor and a nef and big integral polarization form a bounded family, provided that the volume of their sum is bounded from above. As applications, we establish boundedness and effective birationality for adjoint foliated surfaces of general type, as well as a uniform positive lower bound for their volumes. Finally, for ε-lc Calabi–Yau adjoint foliated surfaces, we prove that the volumes of the canonical divisors of rank one foliations on the underlying surfaces belong to a fixed discrete set.

2025

1. JEMS

   Boundedness of polarized log Calabi-Yau fibrations with bounded bases

   Xiaowei Jiang, Junpeng Jiao, and Minzhe Zhu

   Journal of the European Mathematical Society, Jul 2025

   To appear

   Abs arXiv PDF

   We investigate the boundedness problem for log Calabi-Yau fibrations whose bases and general fibers are bounded. We prove that the total spaces of log Calabi-Yau fibrations are bounded in codimension one after fixing some natural invariants. We also prove that the total spaces are bounded if, in addition, the irregularity of the general fibers vanishes. Then we apply our results to the boundedness problem for stable minimal models and fibered Calabi-Yau varieties.

2. IMRN

   Boundedness of stable minimal models with klt singularities

   Minzhe Zhu

   International Mathematics Research Notices, Jan 2025

   Abs DOI PDF

   We investigate the singularities and boundedness of a special kind of algebraic varieties so-called stable minimal models, which are constructed and studied by Birkar. Given a klt stable minimal model with bounded relative volume, if we fix the dimension, Iitaka volume, and a DCC set controlling coefficients, then we show that the singularities of the klt stable minimal model can be controlled uniformly. Furthermore, we prove that with certain bounded data, stable minimal models with klt singularities form a bounded family.

2023

1. Preprint

   On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

   Minzhe Zhu

   Jan 2023

   arXiv:2301.07462

   Abs arXiv

   Given a natural number l and a weak Fano n-fold X with dim\phi_lKX(X)≥n−1, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.

2. Preprint

   On explicit birational geometry for polarised varieties

   Minzhe Zhu

   Mar 2023

   arXiv:2303.12292

   Abs arXiv

   In this paper, we investigate the explicit birational geometry for projective ϵ-lc varieties polarised by nef and big Weil divisors. We show that if X is a projective ϵ-lc variety, H is a nef and big Weil divisor with dim\phi_H(X)≥n−1 and L is an effective Weil divisor such that |L−KX|≠∅ or L−KX is nef, then we can find an explicit lower bound of vol(H) and prove that |L+m′H| is birational for m′≥m, where m is an explicit number which depends only on n and ϵ. This result can be applied to polarised Calabi-Yau varieties, Fano varieties and varieties of general type.