A non-klt counterexample to Shokurov's index conjecture
We give a counterexample to Shokurov's index conjecture for minimal log discrepancies in the non-klt setting.
math.AG↗
arXiv subjects
Yusuke Nakamura
Publications and source records attributed to Yusuke Nakamura.
A characterization of virtually free actions via arc spaces and its application to the lower semi-continuity conjecture
In this paper, we study the precise inversion of adjunction (PIA) conjecture and the lower semi-continuity (LSC) conjecture for hyperquotient singularities. Previously known results for these conjectures in this setting required the singularity to be klt, and without this assumption, a counterexample to the PIA conjecture is known to exist. To resolve this obstacle, we introduce a localized notion of virtually free actions and characterize it via the arc spaces of quotient varieties. Utilizing this characterization, we establish a necessary and sufficient condition for the PIA conjecture to hold for arbitrary hyperquotient singularities, thereby clarifying the mechanism of the counterexample. Furthermore, as an application of this insight, we unconditionally establish the LSC conjecture for arbitrary hyperquotient singularities.
math.AG↗
VIR-Bench: Evaluating Geospatial and Temporal Understanding of MLLMs via Travel Video Itinerary Reconstruction
Recent advances in multimodal large language models (MLLMs) have significantly enhanced video understanding capabilities, opening new possibilities for practical applications. Yet current video benchmarks focus largely on indoor scenes or short-range outdoor activities, leaving the challenges associated with long-distance travel largely unexplored. Mastering extended geospatial-temporal trajectories is critical for next-generation MLLMs, underpinning real-world tasks such as embodied-AI planning and navigation. To bridge this gap, we present VIR-Bench, a novel benchmark consisting of 200 travel videos that frames itinerary reconstruction as a challenging task designed to evaluate and push forward MLLMs' geospatial-temporal intelligence. Experimental results reveal that state-of-the-art MLLMs, including proprietary ones, struggle to achieve high scores, underscoring the difficulty of handling videos that span extended spatial and temporal scales. Moreover, we conduct an in-depth case study in which we develop a prototype travel-planning agent that leverages the insights gained from VIR-Bench. The agent's markedly improved itinerary recommendations verify that our evaluation protocol not only benchmarks models effectively but also translates into concrete performance gains in user-facing applications.
cs.CV↗
Rationality of the multivariate growth series for algebraic sets of virtually abelian groups
We prove the rationality of the multivariate relative growth series for algebraic sets of virtually abelian groups, which had been conjectured by Evetts and Levine.
math.GR↗
Inversion of adjunction for quotient singularities II: Non-linear actions
We prove the precise inversion of adjunction formula for quotient singularities. As an application, we prove the semi-continuity of minimal log discrepancies for hyperquotient singularities. This paper is a continuation of arXiv:2011.07300, and we generalize the previous results to non-linear group actions.
math.AG↗
Ehrhart theory on periodic graphs II: Stratified Ehrhart ring theory
We investigate the "stratified Ehrhart ring theory" for periodic graphs, which gives an algorithm for determining the growth sequences of periodic graphs. The growth sequence $(s_{Γ, x_0, i})_{i \ge 0}$ is defined for a graph $Γ$ and its fixed vertex $x_0$, where $s_{Γ, x_0, i}$ is defined as the number of vertices of $Γ$ at distance $i$ from $x_0$. Although the sequences $(s_{Γ, x_0, i})_{i \ge 0}$ for periodic graphs are known to be of quasi-polynomial type, their determination had not been established, even in dimension two. Our theory and algorithm can be applied to arbitrary periodic graphs of any dimension. As an application of the algorithm, we determine the growth sequences in several new examples.
math.CO↗
On generalized minimal log discrepancy
We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for generalized pairs. We also discuss the theory of complements for generalized pairs.
math.AG↗
Shokurov's index conjecture for quotient singularities
We prove Shokurov's index conjecture for quotient singularities.
math.AG↗
A counterexample to the PIA conjecture for minimal log discrepancies
We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.
math.AG↗
Ehrhart theory on periodic graphs
The purpose of this paper is to extend the scope of the Ehrhart theory to periodic graphs. We give sufficient conditions for the growth sequences of periodic graphs to be a quasi-polynomial and to satisfy the reciprocity laws. Furthermore, we apply our theory to determine the growth series in several new examples.
math.CO↗
Inversion of adjunction for quotient singularities III: semi-invariant case
We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for them. These results extend the results in our first paper, where we prove the same results for complete intersection varieties defined by ``invariant equations".
math.AG↗
Upper bounds of orders of automorphism groups of leafless metric graphs
We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give all metric graphs which have the maximum numbers of automorphisms. The proof is written in terms of graph theory.
math.CO↗
Inversion of adjunction for quotient singularities
We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.
math.AG↗
Diversity in immunogenomics: the value and the challenge
With the advent of high-throughput sequencing technologies, the fields of immunogenomics and adaptive immune receptor repertoire research are facing both opportunities and challenges. Adaptive immune receptor repertoire sequencing (AIRR-seq) has become an increasingly important tool to characterize T and B cell responses in settings of interest. However, the majority of AIRR-seq studies conducted so far were performed in individuals of European ancestry, restricting the ability to identify variation in human adaptive immune responses across populations and limiting their applications. As AIRR-seq studies depend on the ability to assign VDJ sequence reads to the correct germline gene segments, efforts to characterize the genomic loci that encode adaptive immune receptor genes in different populations are urgently needed. The availability of comprehensive germline gene databases and further applications of AIRR-seq studies to individuals of non-European ancestry will substantially enhance our understanding of human adaptive immune responses, promote the development of effective diagnostics and treatments, and eventually advance precision medicine.
q-bio.GN↗
General Relativistic Effects on Hill Stability of Multi-Body Systems I: Stability of Three-Body Systems Containing a Massive Black Hole
We study the effects of general relativistic gravity on the Hill stability, that is, the stability of a multi-body system against a close approach of one orbit to another, which has been hitherto studied mainly in Newtonian mechanics and applied to planetary systems. We focus in this paper on the three-body problem and extend the Newtonian analyses to the general relativistic regime in the post-Newtonian approximation. The approximate sufficient condition for the relativistic Hill stability of three-body systems is derived analytically and its validity and usefulness are confirmed numerically. In fact, relativity makes the system more unstable than Newtonian mechanics in the sense of the Hill stability as expected by our theoretical prediction. The criterion will be useful to analyze the results of large-scale N-body simulations of dense environments, in which the stability of three-body sub-systems is important.
gr-qc↗
A Witt Nadel vanishing theorem for threefolds
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_q$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.
math.AG↗
Dual complex of log Fano pairs and its application to Witt vector cohomology
We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro-Fujino type and a rational point formula in dimension three.
math.AG↗
Minimal model program for log canonical threefolds in positive characteristic
Given a three-dimensional projective log canonical pair over a perfect field of characteristic larger than five, there exists a minimal model program that terminates after finitely many steps.
math.AG↗