Article
Mathematics
Carlos Florentino, Jaime Silva
Summary: The mixed Hodge-Deligne polynomials on complex quasi-projective F-varieties were studied and calculated for varieties with simple mixed Hodge structures. Particularly focused on the case of the maximal torus of an affine reductive group and its Weyl group, explicit formulas for G-character varieties of free abelian groups were obtained as an application, with concrete expressions derived for GL(n, C) and SL(n, C) using partition combinatorics.
Article
Mathematics
Carlos Florentino, Azizeh Nozad, Alfonso Zamora
Summary: In this paper, we generalize a formula of Mozgovoy-Reineke by providing a concrete relation, in terms of plethystic functions, between the generating series for E-polynomials of X_Gamma_G and X_Gamma_irrG. We prove this relation using a natural stratification of X_Gamma_G, combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products. We also adapt our method to the Cartan brane in the moduli space of Higgs bundles. Furthermore, combining our methods with arithmetic ones allows us to obtain explicit expressions for the E-polynomials and Euler characteristics of the irreducible stratum of GL(n,C)-character varieties of some groups Gamma, for low values of n.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
Angel Gonzalez-Prieto
Summary: In this paper, a weak version of quotient for the algebraic action of a group on a variety, called pseudo-quotient, is proposed. Pseudo-quotients focus on the purely topological properties of good Geometric Invariant Theory (GIT) quotients. They provide more flexibility in geometric constructions compared to classical GIT quotients. The interplay between pseudo-quotients and good quotients is investigated, and it is shown that pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties in characteristic zero. As an application, the virtual class of SL2(k)-character varieties for free groups and surface groups, as well as their parabolic counterparts with punctures of Jordan type, is computed.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics, Applied
Carlos Florentino, Azizeh Nozad, Alfonso Zamora
Summary: This study uses geometric methods to prove the equality of E-polynomials for specific complex reductive groups, settling a conjecture by Lawton and Munoz. The proof involves stratification by polystable type and demonstrates the equality of E-polynomials across different strata, particularly in the irreducible strata.
JOURNAL OF GEOMETRY AND PHYSICS
(2021)
Article
Mathematics
Eva Elduque, Moises Herradon Cueto, Laurentiu Maxim, Botong Wang
Summary: In this article, we introduce a method that associates a complex algebraic variety with a morphism to a complex affine torus and defines a natural mixed Hodge structure on the corresponding multivariable cohomological Alexander modules. By applying this method, we prove the quasi-unipotence of monodromy, obtain upper bounds on the sizes of the Jordan blocks of monodromy, and explore the change in the Alexander modules after removing fibers of the map. We also provide an example of a variety whose Alexander module has non-semisimple torsion.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Alexander Petrov
Summary: The authors provide examples of smooth proper rigid-analytic varieties with formal models having projective special fibers that violate Hodge symmetry for cohomology in degrees >= 3, thereby giving a negative answer to a question posed by Hansen and Li.
COMPOSITIO MATHEMATICA
(2021)
Article
Mathematics
Dionisio Peralta, Yamilet Quintana, Shahid Ahmad Wani
Summary: This paper investigates a novel family of mixed-type hypergeometric Bernoulli-Gegenbauer polynomials, exploring their algebraic and differential properties, and their relationships with hypergeometric Bernoulli polynomials. It is found that these polynomials do not fulfill Hanh or Appell conditions.
Article
Mathematics
Salman Abdulali
Summary: We complete the classification of Kuga fiber varieties by demonstrating that a multiple of a representation rho, satisfying Satake's necessary conditions, defines a Kuga fiber variety.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Zili Zhang
Summary: By studying cluster varieties, new examples of the P = W identities of de Cataldo-Hausel-Migliorini are discovered. It is proven that the weight filtration of 2D cluster varieties corresponds to the perverse filtration of elliptic fibrations that are equivalent to certain types of elliptic fibrations. These examples do not originate from character varieties or moduli of Higgs bundles.
MATHEMATICAL RESEARCH LETTERS
(2021)
Article
Mathematics
Paolo Dolce, Roberto Gualdi
Summary: The paper presents a formula relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety X with the Mordell-Weil rank of the Albanese variety of X-K and the rank of the Neron-Seven group of X-K. Additionally, it proves that the numerically trivial arithmetic R-divisors on X are exactly the linear combinations of principal ones.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2022)
Article
Mathematics, Applied
Jian Xiao
Summary: Inspired by previous work on Hodge-index type theorems, this study presents a mixed Hodge-Riemann bilinear relation using the notion of m-positivity. The proof is an adaptation of previous works and it holds with respect to mixed polarizations, some of which satisfy particular positivity conditions but may be degenerate along certain directions. This relation is particularly applicable to fibrations of compact Kahler manifolds.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Mathematics
Takahiro Saito
Summary: In this paper, we study the decomposition of monodromic D-modules and mixed Hodge modules on a smooth algebraic variety, as well as the equivalence between the category of monodromic mixed Hodge modules and the category of gluing data. We also provide a mixed Hodge module structure for the Fourier-Laplace transformation of the underlying D-module of a monodromic mixed Hodge module.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2022)
Article
Mathematics, Applied
Kevin McGerty, Thomas Nevins
Summary: This paper discusses the multiplicative quiver variety associated with a quiver, and proves that its pure cohomology is generated by tautological characteristic classes. Particularly, the pure cohomology of genus g twisted character varieties of GL(n) is also generated by tautological classes.
SELECTA MATHEMATICA-NEW SERIES
(2021)
Article
Mathematics
Stefan Reppen
Summary: This article studies the geometric special fiber of a Hilbert modular variety associated to a totally real field, at a prime unramified in the field. The authors show that the order of vanishing of the Hasse invariant on the fiber is equal to the largest integer m such that the smallest piece of the conjugate filtration lies in the mth piece of the Hodge filtration, which is analogous to Ogus' result on families of Calabi-Yau varieties in positive characteristic. They also demonstrate that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Benjamin Bakker, Yohan Brunebarbe, Jacob Tsimerman
Summary: The research proves a mixed version of Griffiths' conjecture, that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. The proof heavily relies on o-minimality and recent work with B. Klingler.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2023)