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Guodu Chen

Publications and source records attributed to Guodu Chen.

13 recordsLinked to original sources
arXiv2024

Boundedness of complements for log Calabi-Yau threefolds

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that there exists a finite set of positive integers $\mathcal{N}$, such that if a threefold pair $(X/Z\ni z,B)$ has an $\mathbb{R}$-complement which is klt over a neighborhood of $z$, then it has an $n$-complement for some $n\in\mathcal{N}$. We also show the boundedness of complements for $\mathbb{R}$-complementary surface pairs.

math.AG↗
arXiv2024

On the Iitaka volumes of log canonical surfaces and threefolds

Given positive integers $d\geq\kappa$, and a subset $\Gamma\subset [0,1]$, let $\mathrm{Ivol}_{\mathrm{lc}}^{\Gamma}(d,\kappa)$ denote the set of Iitaka volumes of $d$-dimensional projective log canonical pairs $(X, B)$ such that the Iitaka--Kodaira dimension $\kappa(K_X+B)=\kappa$ and the coefficients of $B$ come from $\Gamma$. In this paper, we show that, if $\Gamma$ satisfies the descending chain condition, then so does $\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ for $d\leq 3$. In case $d\leq 3$ and $\kappa=1$, $\Gamma$ and $\mathrm{Ivol}_\mathrm{lc}^\Gamma(d,\kappa)$ are shown to share more topological properties, such as closedness in $\mathbb{R}$ and local finiteness of accumulation complexity. In higher dimensions, we show that the set of Iitaka volumes for $d$-dimensional klt pairs with Iitaka dimension $\geq d-2$ satisfies the DCC, partially confirming a conjecture of Zhan Li. We give a more detailed description of the sets of Iitaka volumes for the following classes of projective log canonical surfaces: (1) smooth properly elliptic surfaces, (2) projective log canonical surfaces with coefficients from $\{0\}$ or $\{0,1\}$. In particular, the minima as well as the minimal accumulation points are found in these cases.

math.AG↗
arXiv2023

Minimal model program for algebraically integrable foliations and generalized pairs

Using techniques from the theory of foliations, we establish the cone theorem and the contraction theorem for lc generalized pairs in full generality, and meanwhile develop the minimal model program for $\mathbb Q$-factorial foliated dlt algebraically integrable foliations. As an application, we obtain the canonical bundle formula for generalized pairs completely, together with several further consequences, including answering a question of Cascini and Spicer.

math.AG↗
arXiv2023

On effective log Iitaka fibrations and existence of complements

We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_X$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a general fiber of an Iitaka fibration of $-K_X$. We also formulate the decomposable Iitaka fibration conjecture, a variation of the effective log Iitaka fibration conjecture which is closely related to the structure of ample models of pairs with non-rational coefficients, and study its relationship with the forestated conjectures.

math.AG↗
arXiv2022

On multiplicities of fibers of Fano fibrations

In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano fibrations over curves.

math.AG↗
arXiv2022

Uniform rational polytopes for Iitaka dimensions

In this paper, we continue to develop the theories on functional pairs and uniform rational polytopes. We show that there is a uniform perturbation for Iitaka dimensions of pseudo-effective lc pairs of fixed dimension with DCC coefficients assuming the non-vanishing conjecture. We also show the existence of uniform rational polytopes for Iitaka dimensions of pseudo-effective lc pairs assuming the non-vanishing conjecture.

math.AG↗
arXiv2022

Degeneration of log Calabi-Yau pairs via log canonical places

Let $(X, \Delta)$ be a projective log canonical Calabi-Yau pair and $L$ an ample $\mathbb{Q}$-line bundle on $X$, we show that there is a correspondence between lc places of $(X, \Delta)$ and weakly special test configurations of $(X, \Delta;L)$.

math.AG↗
arXiv2020

On the termination of flips for log canonical generalized pairs

We prove the termination of flips for 4-dimensional pseudo-effective NQC log canonical generalized pairs. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

math.AG↗
arXiv2020

Boundedness of ($ε, n$)-Complements for projective generalized pairs of Fano type

We show the existence of $(ε,n)$-complements for $(ε,\Rr)$-complementary projective generalized pairs of Fano type $(X,B+M)$ when either the coefficients of $B$ and $μ_j$ belong to a finite set or the coefficients of $B$ belong to a DCC set and $M'\equiv 0$, where $M'=\sumμ_jM_j'$ and $M_j'$ are b-Cartier nef divisors.

math.AG↗