arXiv subjects

Mohammad Farajzadeh-Tehrani

Publications and source records attributed to Mohammad Farajzadeh-Tehrani.

8 recordsLinked to original sources

arXiv2026

On compactifications of the SL(2,C) character varieties of punctured surfaces

This paper addresses some conjectures and questions regarding the absolute and relative compactifications of the $\SL(2,\C)$-character variety of an $n$-punctured Riemann surface without boundary. We study a class of projective compactifications determined by ideal triangulations of the surface and prove explicit results concerning the boundary divisors of these compactifications. Notably, we establish that the boundary divisors are toric varieties and confirm a well-known conjecture asserting that the (dual) boundary complex of any (positive dimensional) relative character variety is a sphere. In a different vein, we enhance and streamline Komyo's compactification method, which utilizes a projective compactification of $\SL(2,\C)$ to compactify the (relative) character varieties. Specifically, we construct a uniform relative compactification over the base space of $\C^n$ and determine its monodromy, addressing a question posed by Simpson.

math.AG↗

arXiv2026

The Geometric P=W conjecture and Thurston's compactification

In this paper, we use new results together with established facts about Thurston's compactification of Teichmüller space to address the geometric P=W conjecture for $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-character variety of any closed surface of genus $g>1$, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus $g$ into $\mathbb{N}^{9g-9}$.

math.GT↗

arXiv2025

What is a stable log map?

Let $X$ be a smooth projective variety over $\mathbb{C}$ with a simple normal crossings divisor $D\subset X$. We compare the notions of stable log maps to $(X,D)$ in algebraic geometry and symplectic topology. In particular, we prove an equivalence between fine (basic) algebraic log maps and symplectic log maps, and we define the symplectic analogue of fine saturated algebraic log maps by refining the notion of log Gromov convergence.

math.AG↗

arXiv2024

The Kauffman Bracket Skein Module at an irreducible representation

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by $4$, is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module.

math.GT↗

arXiv2022

BPS invariants of symplectic log Calabi-Yau fourfolds

Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstruction spaces for maps that have components in a destabilizing (or rubber) component of the target. Finally, we study a higher genus version of these invariants and explain a decomposition of genus one invariants into different contributions.

math.SG↗

arXiv2020

Relative Seiberg-Witten invariants and a sum formula

We study relative Seiberg-Witten moduli spaces and define relative invariants for a pair $(X,Σ)$ consisting of a smooth, closed, oriented 4-manifold $X$ and a smooth, closed, oriented 2-dimensional submanifold $Σ\!\subset\!X$ with positive genus. These relative Seiberg-Witten invariants are meant to be the counterparts of relative Gromov-Witten invariants. We also obtain a sum formula (aka a product formula) that relates the SW invariants of a sum $X$ of two closed oriented 4-manifolds $X_1$ and $X_2$ along a common oriented surface $Σ$ with dual self-intersections to the relative SW invariants of $(X_1,Σ)$ and $(X_2,Σ)$. Our formula generalizes Morgan-Szabó-Taubes' product formula.

math.DG↗

arXiv2020

Pseudoholomorphic Curves Relative to a Normal Crossings Symplectic Divisor: Compactification

Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.

math.SG↗

arXiv2019

Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan-Tian Perturbations

In a previous paper [FT1], for any logarithmic symplectic pair (X,D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural set up for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov-Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov- Witten invariants of arbitrary such pair (X, D).

math.SG↗