arXiv subjects

Mohammad Reza Rahmati

Publications and source records attributed to Mohammad Reza Rahmati.

At least 19 recordsLinked to original sources

arXiv2025

Mixed Hodge Structures on Generalized Theta Divisors and Graph Motives

This article studies the mixed Hodge structures that appear on the complements of generalized theta divisors inside generalized Jacobians of curves with modulus. For a smooth or nodal curve with an effective modulus, the generalized Jacobian is a semiabelian variety, and its generalized theta divisor has a natural determinantal description. Using a smooth compactification with simple normal crossing boundary together with Deligne's theory of logarithmic mixed Hodge complexes, we obtain explicit formulas for the weight filtration on the cohomology of the complement of the generalized theta divisor. The graded pieces of the weight filtration are described in terms of the cohomology of strata determined by the dual graph of the curve and the combinatorics of the modulus. Several examples are worked out, including the nodal cubic and low genus singular curves, showing cases of mixed Tate type as well as more complicated weight behavior. Motivated by similarities with the structure of graph hypersurfaces, we also suggest a conjectural connection between these mixed Hodge structures and motives associated to decorated dual graphs related to Feynman integrals.

math.AG↗

arXiv2025

Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence

We construct a canonical sesquilinear pairing on the relative crystalline cohomology of a smooth proper family of varieties over a complete discretely valued $p$-adic field. Motivated by the role of Saito's higher residue pairing in the theory of primitive forms and complex variations of Hodge structure, we develop a $p$-adic analogue based on the twisted relative de~Rham--Witt complex. We show that this twisted complex defines a filtered $F$-isocrystal whose cohomology carries a natural flat, Frobenius-compatible, and non-degenerate bilinear form. Its specialization at the uniformizer recovers the classical Grothendieck residue on the special fiber, providing a direct bridge between crystalline geometry and residue theory. Using the $p$-adic Riemann--Hilbert correspondence of Faltings and Liu--Zhu, we further identify the resulting pairing with the unique flat extension of this residue form to the corresponding $p$-adic local system. The construction is functorial in the family and compatible with base change and $p$-adic comparison isomorphisms. This yields a genuine $p$-adic analogue of Saito's higher residue pairing and supplies foundational ingredients for a prospective theory of $p$-adic primitive forms, $p$-adic TERP structures, and $p$-adic Frobenius manifolds.

math.AG↗

arXiv2025

Toroidal Compactifications and Stacky Cohomology of Mumford--Tate Domains

Mumford--Tate domains parametrize polarized Hodge structures with fixed Mumford--Tate group and play a central role in the geometry of period maps. Their degenerations are governed by nilpotent orbits and limiting mixed Hodge structures, whose asymptotics are encoded in the logarithmic compactifications of Kato--Usui. In this paper we construct and study the \emph{log--toric Hodge stack} \[ \cD^{\log}_{\MT,Σ} := [D_{\MT,Σ}/Γ], \] obtained from a Mumford--Tate domain $\DM$ and a fan $Σ$ of nilpotent cones by forming the quotient of the Kato--Usui partial compactification $D_{\MT,Σ}$ by a neat arithmetic group $Γ\subset \MT(\Q)$. We show that $\cD^{\log}_{\MT,Σ}$ is a global quotient Deligne--Mumford stack, that it admits a natural logarithmic structure extending the period domain, and that near every boundary stratum associated to a cone $σ\inΣ$ it admits a canonical analytic log--étale chart of the form \[ \bigl([F_σ/G_σ]\times \cT_σ\bigr)^\circ, \] where $F_σ$ is the space of nilpotent orbits modulo unipotent actions, $G_σ$ is a finite symmetry group of the associated limiting mixed Hodge structures, and $\cT_σ$ is a toric Deligne--Mumford stack refining the toric variety $D_σ$ attached to $σ$. This decomposition cleanly separates Hodge-theoretic information from the combinatorial and stacky boundary data.

math.AG↗

arXiv2025

Finite Generation and Structure of Invariant Jets under Non-Reductive Reparametrization

We study invariant jet differentials in the framework of complex hyperbolicity, focusing on the algebra of invariants for the non--reductive reparametrization group $G_k = \mathbb{C}^{\ast} \ltimes U_k$. The paper develops a uniform, representation--theoretic, and graded--algebraic strategy for the $ρ$--action of $G_k$ on $J_k\mathbb{C}^n$, establishing in particular the finite generation of the invariant jet algebra central to the Green--Griffiths--Demailly program. Specifically, we prove that the $\mathbb{C}^{\ast}$--graded algebra of unipotent invariants $\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ is finitely generated for all $n,k$; equivalently, the fiber ring of invariant jet differentials is a finitely generated positively graded $\mathbb{C}$--algebra, so that its projective spectrum $\operatorname{Proj}\,\mathbb{C}[J_k\mathbb{C}^n]^{U_k}$ exists and coincides with the Demailly--Semple tower.

math.RT↗

arXiv2025

Schur-Weyl Duality and Higher Abel-Jacobi Invariants for Tautological Cycles in $\mathcal{M}_{g,n}$

This article investigates the Hodge theory of the moduli space of genus $g$ curves with $n$ marked points, establishing new connections between Schur-Weyl duality for $\mathfrak{sp}_{g}$ and higher Abel-Jacobi invariants. We develop a represe\\ ntation-theoretic framework that decomposes higher Abel-Jacobi invariants of tautological cycles in $C_{g}^{n}$ according to symplectic Lie algebra representations, leveraging the Leray filtration and motivic decompositions compatible with $\mathfrak{sp}_{2g}$-actions. Central to this work is the introduction of \textbf{higher Faber-Pandharipande cycles} $FP_n = π_1^{\times 2}(Δ_{12}^n \cdot ψ_1)$ in $CH^{n+1}(C_g^2)$, a new family of tautological cycles generalizing classical constructions. We prove these cycles are non-torsion under optimal genus constraints: for families over $(n-1)$-dimensional bases, $FP_n$ is not rationally equivalent to zero when $g \geq 3n+1$. Furthermore, we determine the precise position of $FP_n$ in the Leray filtration of $C_g^2 \to M_g$, showing it lies in depth $n+1$ but no deeper, with explicit non-vanishing in $H^{n+1}(M_g, R^{n+1}f_*\mathbb{Q})$ on the $V_{(n+1,1)}$-isotypic component. This yields the first systematic link between Schur-Weyl duality and higher transcendental invariants, revealing that higher diagonals encode geometric phenomena invisible to standard tautological classes.

math.AG↗

arXiv2024

Invariant Jet differentials and Asymptotic Serre duality

We generalize the main result of Demailly \cite{D2} for the bundles $E_{k,m}^{GG}(V^*)$ of jet differentials of order $k$ and weighted degree $m$ to the bundles $E_{k,m}(V^*)$ of the invariant jet differentials of order $k$ and weighted degree $m$. Namely, Theorem 0.5 from \cite{D2} and Theorem 9.3 from \cite{D1} provide a lower bound $\frac{c^k}{k}m^{n+kr-1}$ on the number of the linearly independent holomorphic global sections of $E_{k,m}^{GG} V^* \bigotimes \mathcal{O}(-m δA)$ for some ample divisor $A$. The group $G_k$ of local reparametrizations of $(\mathbb{C},0)$ acts on the $k$-jets by orbits of dimension $k$, so that there is an automatic lower bound $\frac{c^k}{k} m^{n+kr-1}$ on the number of the linearly independent holomorphic global sections of $E_{k,m}V^* \bigotimes \mathcal{O}(-m δA)$. We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. We also prove a Serre duality for asymptotic sections of jet bundles. An application is also given for partial application to the Green-Griffiths conjecture.

math.AG↗

arXiv2024

Nilpotent orbit theorem in $p$-adic Hodge theory

We state and prove three orbit theorems on the period domains for the $p$-adic Hodge structure analogous to the complex case. We shall consider the variation of de Rham (resp. étale) cohomology in a family of projective varieties $f:\mathfrak{X} \to S$ defined over a p-adic field. First, we show that any nilpotent orbit in the period domain of p-adic Hodge structures converges to a semistable point (filtration) in the period domain of the p-adic Hodge structure. Furthermore, the nilpotent orbits of the limit point are asymptotic to the twisted period map [Theorem \ref{thm:nilpotent-orbit}]. The orbit theorems come with some estimates of the distance between the nilpotent orbit and the twisted period map. The distance estimate in the p-adic nilpotent orbit theorem is given concerning the non-archimedean metric and is based on the p-adic Fourier analysis of Amice-Schneider. The result is analogous to the orbit theorems of W. Schmid [\cite{Sch}-1973] on complex Hodge structures. Our proof is based on a \textit{Geometric Invariant Theory} (GIT) criterion for semi-stability (Kempf-Ness theorem) and estimates from the (Amice-Schneider) p-adic Fourier theory. We also state the $SL_2$-orbit theorem in the p-adic case, [Theorem \ref{th:homomorphism}]. Finally, we explain how the nilpotent orbit theorem should be modified and stated for a variation of the mixed Hodge structure [Theorem \ref{thm:mixed-orbit}].}

math.NT↗

arXiv2022

Fock Space of Level infinity and Characters of Vertex Operators

We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality between $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible representations with joint highest weight vector for $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}$. The decomposition of the Fock space $\mathfrak{F}^{\infty}$ into highest weight representations provides a method to calculate and interpret the extended trace.

math.RT↗

arXiv2021

Graded Linearity of Stanley-Reisner Ring of Broken Circuit Complexes

This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring $A=k[x_1, \dots ,x_n]$. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of $A$, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley-Reisner ring of broken circuit complexes and generalize several results of \cite{RV} on the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each strata has a decomposition into uniform matroids. We also present analogs of our results for the Orlik-Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.

math.AC↗

arXiv2021

Geometry of Weighted Homogeneous Spaces

In this paper, we define the weighted homogeneous space (WHS), denoted by $\frac{G}{P}[ψ_H]$ where $ψ_H$ is weight function defined on the set of simple roots of $G$, by an element $H$ in the highest Weyl chamber. The weight function $ψ_H$ describes the action of the maximal torus $T$ on different Bruhat cells and is well behaved via the change of coordinates defined by the action of the Weyl group $W$. The major effort in this text is to prove basic algebraic and geometric properties of a weighted homogeneous space. The definition can be compared with an existing version given by Reid-Corti \cite{CR}. Additionally, we express $\frac{G}{P}[ψ_H]$ as a whole compact quotient of $G/P$ by a certain action of a finite abelian group. Besides, it is presented a criterion when two WHS with possibly different weight systems are isomorphic. The criteria give a simple method to understand the regular maps between two WHS's, defined by matrices with specific polynomial entries. We also explain invariant Kähler differentials on WHS by using certain potential functions on $G$. Our contribution is a generalization of the results presented in \cite{Al, AL, AKQ}. For that, we explain how the weights affect different computations of chern classes of line bundles given in \cite{AL, AKQ}. Finally, we provide a result on the coordinate ring of a WHS by cluster algebras associated to weighted quivers. Specifically, we show that the coordinate ring of a WHS is a weighted cluster algebra of finite type. In this case, the corresponding Dynkin quiver is equipped with a weight function defined on the vertices where the mutations also affect the weights. We present an embedding of a WHS in a product of weighted projective spaces, showing that the coordinate ring is a weighted graded algebra.

math.RT↗

arXiv2021

Calabi-Yau attractor varieties and degeneration of Hodge structure

We study the structure of string theory flux compactification for a general family of elliptic CY 3-folds. We investigate the locus of the attractor points of the flux compactification in type IIB string theory on the boundary components of period domains. Specifically we give equations describing this locus through the asymptotic of nilpotent orbits on period domains. our approach is a mixture of techniques of asymptotic Hodge theory and the numerical period vectors used in physics.

math.AG↗

arXiv2020

A question on generalization of partition functions of CY 3-folds in String Theory

This is an expositoray article on the topological string partition function promoting an extension of the partition function of open Gromov-Witten theory of CY 3-folds defined by the trace of vertex operators. We also give a brief survey of their connection to the theory of Hilbert scheme of points on surface. Specifically; we apply infinitely many Cassimir operators twisted to the vertex operator computing the amplitude. The case of finite number of twists has been well discussed in the mathematics and Physics literature.

physics.gen-ph↗

arXiv2020

Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles

We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical category. They naturally identify the tropical Bloch-Beilinson filtration on the tropical Chow group. We examine this construction on the moduli of tropical curves with marked points, in order to study the tropical tautological classes in the tautological ring of $\mathcal{M}_{g,n}^{\text{trop}}$. The expectation is the nontriviality of these cycles could be examined with less complexity in the tropical category. The construction is compatible with the tropicalization functor on the category of schemes, and the aforementioned procedure will also provide an alternative way to examine the relations in the tautological ring of $\mathcal{M}_{g,n}$ in the schemes category.

math.AG↗

arXiv2020

Hodge-Arakelov inequalities for family of surfaces fibered by curves

The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variations of Hodge structure (cf. \cite{GGK}), in a commutative triangle of fibrations. We have proposed to relate the degrees of the Hodge bundles in the two families. We also compare the Fujita decomposition of Hodge bundles in these fibrations. We examine various identities and relations between Hodge numbers and degrees of the Hodge bundles in different levels.

math.AG↗

arXiv2020

Vertex algebras and Hodge structures

We compare the context of Hodge structures with that of vertex algebras of conformal field theory. Vertex algebras appear as the highest weight representations of infinite dimensional Lie algebras. A correspondence between Higgs bundles and opers already is known as non-abelian Hodge theorem due to C. Simpson. The Beilinson-Bernstein localization (correspondence) also compares the context of variation of Hodge structure with that of highest weight modules over flag manifolds of semisimple Lie groups. A more general analogue of the Bernstein correspondence over a local manifold can also be formulted in the context of geometric Langlands correspondence. We discuss a generalized version of Harish-Chandra modules called Wakimoto modules and a generalized Harish-Chandra homomorphism. This text is mainly an expository discussion with a new insight toward the two concepts. We conclude with an explanation of geometric Langlands correspondence.

math.RT↗

arXiv2020

de Rham Cohomology of Period Domains

This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal compactification of these domains using known formulas on cohomology or Chow rings of toric structures. The text is expository and we have tried to connect some existing ideas that probably their relations not processed in the literature of Hodge theory. We state the significance of ideas as they naturally could be related, probably with not serious mathematical proof. The proofs stated in the text maybe expressed in a more serious context.

math.AG↗

arXiv2020

Curvature of Metrics on Semple Jet bundles

We discuss the Morse estimates for the curvature of several metrics on Semple weighted projective bundle over a projective variety. Following Demailly works on holomorphic Morse inequalities we show an analogue of his results along the Green-Griffiths conjecture for invariant jets.

math.DG↗

arXiv2020

Variation of Hodge Structure and Hodge modules

This text is an expository survey on the interplay between polarized variation of Hodge structure (PVHS) and the formalism of Hodge modules. We specifically review the extensions of a PVMHS over their singularities and its relation to mixed Hodge modules.

math.AG↗