arXiv subjects

Jihao Liu

Publications and source records attributed to Jihao Liu.

At least 37 records · Page 2Linked to original sources

arXiv2026

A counterexample to the near-quadratic Elekes--Rónyai expander conjecture over $\mathbb R$

We disprove the near-quadratic Elekes--Rónyai expander conjecture over $\mathbb R$. The counterexample is a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers on which its image has a fixed power saving from quadratic size. The main result of this paper is obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof relies on a recent construction by OpenAI of an infinite tower of number fields.

math.NT↗

arXiv2026

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↗

arXiv2026

Optimal bend-and-break for foliations

We show that for every foliation $\mathcal{F}$ of rank $r$ on a normal projective variety, the optimal constant in the bend-and-break inequality for tangent rational curves is $r+1$. The proof combines the method of Bogomolov--McQuillan and the bend-and-shatter method developed by Jovinelly--Lehmann--Riedl. The proof of the main result of this paper substantially uses generative AI, particularly the Rethlas system.

math.AG↗

arXiv2026

On a question of Mauri and Moraga

We give negative answers to both parts of a question of Mauri and Moraga on log Calabi-Yau pairs whose boundary decomposes into big divisors. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

math.AG↗

arXiv2026

A question on klt type varieties of Han and Jiang

We prove that being of klt type is not an open condition in flat families of varieties. This answers a question of Han and Jiang. The construction in this paper substantially uses generative AI: the general idea for the counterexample was suggested by ChatGPT Pro 5.5, and the explicit example was found and proved by the Rethlas system.

math.AG↗

arXiv2026

Shokurov's global index conjecture for threefold foliations

We prove Shokurov's global index conjecture for foliations in dimension at most three. This answers a question of the first author, Meng, and Xie in dimension three. The main result of this paper is partially obtained by generative AI, particularly the Rethlas system.

math.AG↗

arXiv2026

Boundedness of total Cartier indices for rational singularities in families

We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the higher-dimensional case separately, was originated by generative AI, particularly the Rethlas system, and was substantially corrected and elaborated by hand.

math.AG↗

arXiv2026

On a question of Kollár and Kovács

We answer a question of Kollár and Kovács by constructing a flat projective morphism to a smooth curve whose fibers are Cohen--Macaulay and reduced, whose generic fiber is smooth, and for which the first cohomology of the structure sheaf of the fibers is not constant. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

math.AG↗

arXiv2026

An example of a very non-movable effective divisor

We give a negative answer to a question of Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa, and Testa on effective divisors of positive self-intersection on smooth projective surfaces. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

math.AG↗

arXiv2026

The minimal volume of stable surfaces of rank one

We determine the minimal volume of a stable surface of rank one, and show that the surface attaining this minimum is unique up to isomorphism. This resolves a conjecture of Alexeev and the second author. Of independent interest, the decisive step of the proof uses a plurigenus inequality re-derived by an AI chatbot and applied as a pluricanonical filter; we further apply this filter to rule out additional cases in the classification of small-volume threefolds of general type, and in Kollár's algebraic Montgomery--Yang problem. The underlying inequality has classical antecedents. To our knowledge this is the first paper in birational geometry to claim a C2-level human--AI collaboration in the sense of Feng et al., where the AI's contribution is the recognition that this inequality functions as the decisive pluricanonical filter in the basket analysis.

math.AG↗

arXiv2026

Birational boundedness of stable families

We prove that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. This is a consequence of a substantially stronger statement, formulated a priori independently of stable families: algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. In this way, the birational geometry of foliations provides a systematic framework for approaching classical boundedness problems for fibrations. A key input is our proof of M\textsuperscript{c}Kernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations. This may be viewed as the foliated analogue of Shokurov's ACC conjecture for log canonical thresholds, proved in the classical setting by Hacon--M\textsuperscript{c}Kernan--Xu. As applications, we establish two boundedness criteria for Fano algebraically integrable adjoint foliated structures: Birkar's criterion for exceptional Fanos, and Jiang's criterion for Fanos for which both Tian's $α$-invariant and the anti-canonical volume are bounded away from zero. We also obtain several results on the birational geometry of algebraically integrable adjoint foliated structures, including lower bounds for adjoint volumes, boundedness of automorphism groups, and ACC theorems for pseudo-effective thresholds, $\mathbb{R}$-complementary thresholds, and the Fano spectrum.

math.AG↗

arXiv2026

Seedance 2.0: Advancing Video Generation for World Complexity

Seedance 2.0 is a new native multi-modal audio-video generation model, officially released in China in early February 2026. Compared with its predecessors, Seedance 1.0 and 1.5 Pro, Seedance 2.0 adopts a unified, highly efficient, and large-scale architecture for multi-modal audio-video joint generation. This allows it to support four input modalities: text, image, audio, and video, by integrating one of the most comprehensive suites of multi-modal content reference and editing capabilities available in the industry to date. It delivers substantial, well-rounded improvements across all key sub-dimensions of video and audio generation. In both expert evaluations and public user tests, the model has demonstrated performance on par with the leading levels in the field. Seedance 2.0 supports direct generation of audio-video content with durations ranging from 4 to 15 seconds, with native output resolutions of 480p and 720p. For multi-modal inputs as reference, its current open platform supports up to 3 video clips, 9 images, and 3 audio clips. In addition, we provide Seedance 2.0 Fast version, an accelerated variant of Seedance 2.0 designed to boost generation speed for low-latency scenarios. Seedance 2.0 has delivered significant improvements to its foundational generation capabilities and multi-modal generation performance, bringing an enhanced creative experience for end users.

cs.CV↗

arXiv2026

Existence of the minimal model program for log canonical generalized pairs

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and Kollár-type gluing theory, we prove the existence of flips for log canonical generalized pairs without assuming the klt condition, the NQC condition, or $\mathbb Q$-factoriality. Together with the cone and contraction theorems, this yields the existence of the minimal model program for arbitrary log canonical generalized pairs.

math.AG↗

arXiv2026

Dual complexes of qdlt Fano type models and strong complete regularity

We introduce birational strong complete regularity and strong complete regularity, two numerical invariants for pairs of (relative) Fano type. They are defined using variants of qdlt Fano type models and the dimension of the dual complex of the reduced boundary, and can be viewed as Fano type refinements of Shokurov's complete regularity. We establish basic properties of these invariants and clarify its relation to models of qdlt Fano type appearing in K-stability. In particular, we prove that any pair with maximal birational strong complete regularity is $1$-complementary, and the thresholds where birational strong complete regularity or strong complete regularity jumps satisfy the ascending chain condition.

math.AG↗

arXiv2026

The minimal volume of surfaces of log general type with non-empty non-klt locus

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. Furthermore, the ample model $V$ achieving the minimal volume is determined uniquely up to isomorphism. The canonical embedding presents $V$ as a degree $86$ hypersurface of $\mathbb P(6,11,25,43)$. This motivates a one-parameter deformation of $V$ to klt stable surfaces within the weighted projective space. Consequently, we identify a $\textit{complete}$ rational curve in the corresponding moduli space $M_{\frac{1}{825}}$. As an important application, we deduce that the smallest accumulation point of the set of volumes for projective log canonical surfaces equals $\frac{1}{825}$.

math.AG↗

arXiv2025

Seedance 1.5 pro: A Native Audio-Visual Joint Generation Foundation Model

Recent strides in video generation have paved the way for unified audio-visual generation. In this work, we present Seedance 1.5 pro, a foundational model engineered specifically for native, joint audio-video generation. Leveraging a dual-branch Diffusion Transformer architecture, the model integrates a cross-modal joint module with a specialized multi-stage data pipeline, achieving exceptional audio-visual synchronization and superior generation quality. To ensure practical utility, we implement meticulous post-training optimizations, including Supervised Fine-Tuning (SFT) on high-quality datasets and Reinforcement Learning from Human Feedback (RLHF) with multi-dimensional reward models. Furthermore, we introduce an acceleration framework that boosts inference speed by over 10X. Seedance 1.5 pro distinguishes itself through precise multilingual and dialect lip-syncing, dynamic cinematic camera control, and enhanced narrative coherence, positioning it as a robust engine for professional-grade content creation. Seedance 1.5 pro is now accessible on Volcano Engine at https://console.volcengine.com/ark/region:ark+cn-beijing/experience/vision?type=GenVideo.

cs.CV↗

arXiv2025

Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures

Assuming the abundance conjecture in dimension $d$, we establish a non-algebraicity criterion of foliations: any log canonical foliation of rank $\le d$ with $ν\neqκ$ is not algebraically integrable, answering question of Ambro--Cascini--Shokurov--Spicer. Under the same hypothesis, we prove abundance for klt algebraically integrable adjoint foliated structures of dimension $\le d$ and show the existence of good minimal models or Mori fiber spaces. In particular, when $d=3$, all these results hold unconditionally. Using similar arguments, we solve a problem proposed by Lu and Wu on abundance of surface adjoint foliated structures that are not necessarily algebraically integrable.

math.AG↗

arXiv2025

Variation of algebraically integrable adjoint foliated structures

Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type, we show the finiteness of ample models if the parameter is sufficiently close to $1$. When the ambient variety is of general type, we show the finiteness of ample models for all parameters. A key ingredient in our proof is the equivalence between the existence of minimal models and the termination of MMP with scaling for algebraically integrable adjoint foliated structures.

math.AG↗