arXiv subjects

Mohammad Farajzadeh Tehrani

Publications and source records attributed to Mohammad Farajzadeh Tehrani.

12 recordsLinked to original sources

arXiv2021

Normal Crossings Singularities for Symplectic Topology: Structures

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures hav applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi's formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor.

math.SG↗

arXiv2020

Limits of stable maps in a semi-stable degeneration

Given a semistable degeneration with a simple normal crossings central fiber, Abramovich-Chen-Gross-Siebert [3] proved a degeneration formula that relates the moduli spaces of stable maps in smooth fibers to certain moduli spaces of log-smooth maps in the central fiber. In this paper, we study the same problem from an analytic point of view. We prove that the limiting stable maps in the central fiber satisfy specific combinatorial and analytical conditions. Furthermore, we explain the deformation-obstruction theory of the moduli spaces arising from these conditions, derive a degeneration formula, and work out an explicit example. The earlier version [8] of this paper contains an outline of these ideas for the symplectic category.

math.SG↗

arXiv2019

Normal Crossings Singularities for Symplectic Topology, II

In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a variety of geometric structures associated with normal crossings singularities in algebraic geometry.

math.SG↗

arXiv2017

Singularities and Semistable Degenerations for Symplectic Topology

We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.

math.SG↗

arXiv2017

Normal Crossings Singularities for Symplectic Topology

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors.

math.SG↗

arXiv2017

The Smoothability of Normal Crossings Symplectic Varieties

Our previous paper introduces topological notions of normal crossings symplectic divisor and variety and establishes that they are equivalent, in a suitable sense, to the desired geometric notions. Friedman's d-semistability condition is well-known to be an obstruction to the smoothability of a normal crossings variety in a one-parameter family with a smooth total space in the algebraic geometry category. We show that the direct analogue of this condition is the only obstruction to such smoothability in the symplectic topology category. Every smooth fiber of the families of smoothings we describe provides a multifold analogue of the now classical (two-fold) symplectic sum construction; we thus establish an old suggestion of Gromov in a strong~form.

math.SG↗

arXiv2017

Normal Crossings Degenerations of Symplectic Manifolds

We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross-Siebert and B. Parker's programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.

math.SG↗

arXiv2017

Gromov-Witten theory via Kuranishi structures

In this expository manuscript, we review the construction of Gromov-Witten virtual fundamental class via FOOO's theory of Kuranishi structures for moduli spaces of pseudo-holomorphic maps defined on closed Riemann surfaces. We consider constraints coming from the ambient space and Deligne-Mumford moduli, called primary insertions, as well as intrinsic classes such as $ψ$-classes and Hodge classes.

math.SG↗

arXiv2016

Notes on genus one real Gromov-Witten invariants

In this paper, we propose a definition of genus one real Gromov-Witten invariants for certain symplectic manifolds with real a structure, including Calabi-Yau threefolds, and use equivariant localization to calculate certain genus one real invariants of the projective space. For this definition, we combine three moduli spaces corresponding to three possible types of involutions on a symplectic torus, by gluing them along common boundaries, to get a moduli space without codimension-one boundary and then study orientation of the total space. Modulo a technical conjectural lemma, we can prove that the result is an invariant of the corresponding real symplectic manifold. In the aforementioned example, our main motivation is to show that the physicists expectation for the existence of separate Annulus, Mobius, and Klein bottle invariants may not always be true.

math.SG↗

arXiv2015

Counting genus zero real curves in symplectic manifolds

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on $\mathbb{P}^{4n-1}$.

math.SG↗

arXiv2013

Automorphism group of Batyrev Calabi-Yau threefolds

In this paper, we will prove that all Batyrev Calabi-Yau threefolds, arising from a small resolution of a generic hyperplane section of a reflexive Fano-Gorenstein fourfold, have finite automorphism group. Together with Morrison conjecture, this suggests that Batyrev Calabi-Yau threefolds should have a polyhedral Kahler (ample) cone.

math.AG↗

arXiv2012

Open GW theory on symplectic manifolds and symplectic cutting

Let $(X,\om)$ be a symplectic manifold and $L$ be a Lagrangian submanifold diffeomorphic to $S^n$, $\R¶^n$, or a Lens space of a certain type. Using the symplectic cut and symplectic sum constructions, we express the open Gromov-Witten invariants of $(X,L)$ in terms of open Gromov-Witten invariants of a pair $(X_-,L)$ determined by $L$ and the standard Gromov-Witten invariants of a symplectic manifold $X_+$ determined by $(X,L)$. We also describe other applications of this approach.

math.SG↗