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Let $\mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $\mathcal{A}$. By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci \cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner \cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga \cite[Example 3.2]{ISY22} gives a negative answer to this question., Comment: 8 pages, 2 figures

Let X be a complex smooth quasi-projective variety with an epimorphism $\nu \colon \pi_1(X)\twoheadrightarrow \mathbb{Z}^n$. We survey recent developments about the asymptotic behaviour of Betti numbers with any field coefficients and the order of the torsion part of singular integral homology of finite abelian covers of $X$ associated to $\nu$, known as the $L^2$-type invariants. We give relations between $L^2$-type invariants, Alexander invariants and cohomology jump loci. When $\nu$ is orbifold effective, we give explicit formulas for $L^2$-invariants at homological degree one in terms of geometric information of $X$. We also propose several related open questions for hyperplane arrangement complement., Comment: 21 pages. There are some sentences in the first version which are not true and misleading. Those are contained in the paragraphs from Theorem 4.7 to Question 4.10 and now revised. arXiv admin note: text overlap with arXiv:2110.03356

In this study, we introduce Orion-14B, a collection of multilingual large language models with 14 billion parameters. We utilize a data scheduling approach to train a foundational model on a diverse corpus of 2.5 trillion tokens, sourced from texts in English, Chinese, Japanese, Korean, and other languages. Additionally, we fine-tuned a series of models tailored for conversational applications and other specific use cases. Our evaluation results demonstrate that Orion-14B achieves state-of-the-art performance across a broad spectrum of tasks. We make the Orion-14B model family and its associated code publicly accessible https://github.com/OrionStarAI/Orion, aiming to inspire future research and practical applications in the field., Comment: Authors are alphabetically listed by last names, except the corresponding author who is listed last

In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams for complexified line arrangement to any complex line arrangement., Comment: 15 pages

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism $\nu\colon\pi_1(X)\twoheadrightarrow H$, where $H$ is a finitely generated abelian group with $\mathrm{rank}H\geq 1$. In this paper, we study the asymptotic behaviour of Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to $\nu$, known as the $L^2$-type invariants. When $\nu$ is orbifold effective, we give explicit formulas of these invariants at degree 1. This generalizes the authors' previous work for $H\cong \Z$., Comment: 12 pages, to appear in Chinese Annals of Mathematics, series B. arXiv admin note: text overlap with arXiv:2110.03356

Using the Brylinski-Radon transformation, we prove that under a generic surjective linear map from C^n to C^m, the pushforward of a perverse sheaf is perverse, modulo shifts of constant sheaves., Comment: 9 pages

Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $\omega \in H^1(X,\mathbb{Z}_2)$. Denote the double covering space by $X^\omega$. There exists a corresponding non-trivial rank one $\mathbb{Z}$-local system $\mathcal{L}_\omega$ on $X$. What is the relation between the integral homology groups of $X^\omega$ and the homology groups of the local system $\mathcal{L}_\omega$? When $X$ is homotopy equivalent to a minimal CW complex, we give a complete answer to this question. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga for hyperplane arrangement complement. As an application, when $X$ is a hyperplane arrangement complement and $\mathcal{L}_\omega$ satisfies certain conditions, we show that $H_*(X^\omega,\mathbb{Z})$ is combinatorially determined., Comment: Final version, to appear in Proc. Amer. Math. Soc

We prove a Cohen-Dimca-Orlik type theorem for rank one $\mathbb{Z}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara., Comment: v2: the paper is completely rewritten, an error in v1 is fixed

We present a benchmark dataset for evaluating method-level code generation task. The benchmark contains a dataset of 175 samples for automated evaluation and a dataset of 161 samples for manual evaluation. We also present a new metric for automatically evaluating the correctness of the generated code, and a set of criteria to manually evaluating the overall quality of the generated code.

Let X be a complex smooth quasi-projective variety with a fixed epimorphism $\nu\colon\pi_1(X)\twoheadrightarrow \mathbb{Z}$. In this paper, we consider the asymptotic behaviour of invariants such as Betti numbers with all possible field coefficients and the order of the torsion part of singular integral homology associated to $\nu$, known as the $L^2$-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of $X$ when $\nu$ is orbifold effective. The proof relies on a study about cohomological degree one jump loci of $X$. We extend part of Arapura's result for cohomological degree one jump loci of $X$ with complex field coefficients to the one with positive characteristic field coefficients. As an application, when $X$ is a hyperplane arrangement complement, a combinatoric upper bound is given for the number of parallel positive dimensional components in cohomological degree one jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu for hyperplane arrangement., Comment: 35 pages.Section 4 are completely rewritten. We give new and shorter proofs for the main results using Hochschild-Serre spectral sequence

We study non-abelian versions of the Mellin transformations, originally introduced by Gabber-Loeser on complex affine tori. Our main result is a generalization to the non-abelian context and with arbitrary coefficients of the t-exactness of Gabber-Loeser's Mellin transformation. As an intermediate step, we obtain vanishing results for the Sabbah specialization functors. Our main application is to construct new examples of duality spaces in the sense of Bieri-Eckmann, generalizing results of Denham-Suciu., Comment: 18 pages, comments are very welcome

Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety $X$; one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This result is related to Kotschick's conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if $X$ has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients. In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of $H^0(X,\Omega_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1., Comment: Final version to appear in Math Zeitschrift: rewrote the introduction and abstract from a slightly different perspective. Shortened exposition, with some deleted proofs as per referee's comments. Added a special case of the main theorem in positive characteristics (Proposition 3.7). 18 pages, comments are very welcome

Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n