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Minzhe Zhu

Publications and source records attributed to Minzhe Zhu.

5 recordsLinked to original sources
arXiv2026

Boundedness of polarized foliated surfaces

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.

math.AG↗
arXiv2026

Boundedness of polarized log Calabi-Yau fibrations with bounded bases

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.

math.AG↗
arXiv2025

Boundedness of stable minimal models with klt singularities

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.

math.AG↗
arXiv2023

On explicit birational geometry for polarised varieties

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 $\dim\overline{φ_{H}(X)}\geq n-1$ and $L$ is an effective Weil divisor such that $|L-K_X|\neq \emptyset$ or $L-K_X$ is nef, then we can find an explicit lower bound of $\text{vol}(H)$ and prove that $|L+m^\prime H|$ is birational for $m^\prime\geq m$, 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, generalizing the results in [CEW22] and [Zhu23].

math.AG↗