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Elham Izadi

Publications and source records attributed to Elham Izadi.

11 recordsLinked to original sources
arXiv2024

Szegő kernels and Scorza quartics on the moduli space of spin curves

We describe an extension at the level of the moduli space of stable spin curves of genus g of the map associating to an ineffective spin structure its Scorza curve (equivalently, the vanishing locus of its Szegő kernel). We compute the class of the Szegő-Hodge bundle, then find an unconditional new interpretation, in terms of theta constants, of the Scorza quartic uniquely associated to an even spin structure. Our results describe the superperiod map from the moduli space of supersymmetric curves in the neighborhood of the theta-null divisor and provide a lower bound for the slope of the movable cone of the moduli space of spin curves.

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arXiv2024

Some density results for hyperk\"ahler manifolds

Lagrangian fibrations of hyperk\"ahler manifolds are induced by semi-ample line bundles which are isotropic with respect to the Beauville-Bogomolov-Fujiki form. For a non-isotrivial family of hyperk\"ahler manifolds over a complex manifold $S$ of positive dimension, we prove that the set of points in $S$, for which there is an isotropic class in the Picard lattice of the corresponding hyperk\"ahler manifold represented as a fiber over that point, is analytically dense in $S$. We also prove the expected openness and density of the locus of polarised hyperk\"ahler manifolds that admit a nef algebraic isotropic line bundle.

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arXiv2022

Hodge classes on the moduli space of W(E_6)-covers and the geometry of A_6

In previous work we showed that the Hurwitz space of W(E_6)-covers of the projective line branched over 24 points dominates via the Prym-Tyurin map the moduli space A_6 of principally polarized abelian 6-folds. Here we determine the 25 Hodge classes on the Hurwitz space of W(E_6)-covers corresponding to the 25 irreducible representations of the Weyl group W(E_6). This result has direct implications to the intersection theory of the toroidal compactification A_6. In the final part of the paper, we present an alternative, elementary proof of our uniformization result on A_6 via Prym-Tyurin varieties of type W(E_6).

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arXiv2021

Hyperkähler manifolds

We give an elementary introduction to hyperkähler manifolds, survey some of their interesting properties and some open problems.

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arXiv2020

Symmetric correspondences with decomposable minimal equation

We study symmetric correspondences with completely decomposable minimal equation on smooth projective curves $C$. The Jacobian of $C$ then decomposes correspondingly. For all positive integers $g$ and $\ell$, we give series of examples of smooth curves $C$ of genus $n^\ell (g-1) +1$ with correspondences satisfying minimal equations of degree $\ell+1$ such that the Jacobian of $C$ has at least $2^\ell$ isogeny components.

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arXiv2020

Twistor lines in the period domain of complex tori

As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of $Compl$, of degree $2n$ in the Plücker embedding of $Compl$.

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arXiv2019

Surfaces generating the even primal cohomology of an abelian fivefold

Given a very general abelian fivefold $A$ and a principal polarization $Θ\subset A$, we construct surfaces generating the algebraic part of the middle cohomology $H^4(Θ, {\mathbb Q})$, and determine the intersection pairing between these surfaces. In particular, we obtain a new proof of the Hodge conjecture for $H^4(Θ, {\mathbb Q})$ and show that it contains a copy of the root lattice of $E_6$.

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arXiv2018

The uniformization of the moduli space of principally polarized abelian 6-folds

Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym-Tyurin map from the Hurwitz space to A_6 in the terms of syzygies of the Abel-Prym-Tyurin curve.

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arXiv2018

The irreducible components of the primal cohomology of the theta divisor of an abelian fivefold

The primal cohomology $\mathbb{K}_\mathbb{Q}$ of the theta divisor $Θ$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\mathbb{K}_\mathbb{Q}^{+1}$, resp. $\mathbb{K}_\mathbb{Q}^{-1}$ under the action of $-1$. For smooth $Θ$, these have dimension $6$ and $72$ respectively. We show that $\mathbb{K}_\mathbb{Q}^{+1}$ consists of Hodge classes and, for a very general ppav, $\mathbb{K}_\mathbb{Q}^{-1}$ is a simple Hodge structure of level $2$.

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arXiv2000

Half twists and the cohomology of hypersurfaces

A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k-1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondance for certain cubic 4-folds.

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