arXiv subjects

Alessandro Ghigi

Publications and source records attributed to Alessandro Ghigi.

At least 19 recordsLinked to original sources

arXiv2025

Prym varieties and projective structures on Riemann surfaces

Given an étale double covering $π\, :\, \widetilde{C}\, \longrightarrow\, C$ of compact Riemannsurfaces with $C$ of genus at least two, we use the Prym variety of the cover to construct canonical projective structures on both $\widetilde C$ and $C$. This construction can be interpreted as a section of an affine bundle over the moduli space of étale double covers. The $\overline{\partial}$--derivative of this section is a (1,1)--form on the moduli space. We compute this derivative in terms of Thetanullwert maps. Using the Schottky--Jung identities we show that, in general, the projective structure on $C$ depends on the cover.

math.AG↗

arXiv2024

Theta functions and projective structures

Given a compact Riemann surface $X$, we consider the line, in the space of sections of $2Θ$ on $J^0(X)$, orthogonal to all the sections that vanish at the origin. This line produces a natural meromorphic bidifferential on $X\times X$ with a pole of order two on the diagonal. This bidifferential is extensively investigated. In particular we show that it produces a projective structure on $X$ which is different from the standard ones.

math.AG↗

arXiv2024

Theta bundle, Quillen connection and the Hodge theoretic projective structure

There are two canonical projective structures on any compact Riemann surface of genus at least two: one coming from the uniformization theorem, and the other from Hodge theory. They produce two (different) families of projective structures over the moduli space $M_g$ of compact Riemann surfaces. A recent work of Biswas, Favale, Pirola, and Torelli shows that families of projective structures over $M_g$ admit an equivalent characterization in terms of complex connections on the dual $\mathcal{L}$ of the determinant of the Hodge line bundle over $M_g$; the same work gave the connection on $\mathcal L$ corresponding to the projective structures coming from uniformization. Here we construct the connection on $\mathcal L$ corresponding to the family of Hodge theoretic projective structures. This connection is described in three different ways: firstly as the connection induced on $\mathcal L$ by the Chern connection of the $L^2$-metric on the Hodge bundle, secondly as an appropriate root of the Quillen metric induced by the (square of the) Theta line bundle on the universal family of abelian varieties, endowed with the natural Hermitian metric given by the polarization, and finally as Quillen connection gotten using the Arakelov metric on the universal curve, modified by Faltings' delta invariant.

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arXiv2023

Topological types of actions on curves

We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface $C$ of genus at least $g \geq 2$ with $C/G \cong \mathbb{P}^1$.

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arXiv2023

A topological construction of families of Galois covers of the line

We describe a new construction of families of Galois coverings of the line using basic properties of configuration spaces, covering theory, and the Grauert-Remmert Extension Theorem. Our construction provides an alternative to a previous construction due to González-Díez and Harvey (which uses Teichmüller theory and Fuchsian groups) and, in the case the Galois group is non-abelian, corrects an inaccuracy therein.

math.AG↗

arXiv2021

Some evidence for the Coleman-Oort conjecture

The Coleman-Oort conjecture says that for large $g$ there are no positive-dimensional Shimura subvarieties of $\mathsf{A}_g$ generically contained in the Jacobian locus. Counterexamples are known for $g\leq 7$. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: a) the Galois group is cyclic, b) it is abelian and the family is 1-dimensional, and c) $g\leq 9$. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for $g\leq 100$ there are no other families than those already known.

math.AG↗

arXiv2021

Bergman kernel and period map for curves

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative - in a suitable sense - of this meromorphic form.

math.AG↗

arXiv2020

Infinitely many Shimura varieties in the Jacobian locus for $g \leq 4$

We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.

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arXiv2019

Totally geodesic subvarieties in the moduli space of curves

In this paper we study totally geodesic subvarieties $Y \subset \mathsf{A}_g$ of the moduli space of principally polarized abelian varieties with respect to the Siegel metric, for $g\geq 4$. We prove that if $Y$ is generically contained in the Torelli locus, then $\dim Y \leq (7g -2)/3$.

math.AG↗

arXiv2019

Meromorphic limits of automorphisms

Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probability measures on $X$. In particular we obtain an extension of Furstenberg lemma to manifolds in the class $\mathscr{C}$.

math.CV↗

arXiv2018

Remarks on the abelian convexity theorem

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.

math.DG↗

arXiv2018

Fujita decomposition and Hodge loci

This paper contains two results on Hodge loci in the moduli space of curves. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fibers is contained in a proper Hodge locus. The second result deals with divisors in the moduli space of curves. It is proved that the image of a divisor in the moduli of principally polarized abelian varieties is not contained in a proper totally geodesic subvariety. It follows that a Hodge locus in the moduli space of curves has codimension at least 2.

math.AG↗

arXiv2018

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.

math.AG↗

arXiv2018

Shimura curves in the Prym locus

We study Shimura curves of PEL type in $\mathsf{A}_g$ generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is $\mathbb{P}^1$. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples gotten in this way up to genus 28. We obtain 43 Shimura curves generically contained in the unramified Prym locus and 9 families generically contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

math.AG↗

arXiv2016

Stability of measures on Kähler manifolds

Let $(M,ω)$ be a Kähler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of $K^\mathbb{C}$ on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied by Hersch and Bourguignon-Li-Yau.

math.DG↗

arXiv2015

On totally geodesic submanifolds in the Jacobian locus

We study submanifolds of A_g that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] in M_g in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of A_g obtained from cyclic covers of the projective line. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.

math.AG↗

arXiv2014

Shimura varieties in the Torelli locus via Galois coverings

Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples found by Moonen-Oort. We get 7 new non-abelian examples.

math.AG↗

arXiv2014

Invariant convex sets in polar representations

We study a compact invariant convex set $E$ in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of $K$ on $\mathfrak{p}$, where $K$ is a maximal compact subgroup of a real semisimple Lie group $G$ with Lie algebra $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. If $\mathfrak{a} \subset \mathfrak{p}$ is a maximal abelian subalgebra, then $P=E\cap \mathfrak{a}$ is a convex set in $\mathfrak{a}$. We prove that up to conjugacy the face structure of $E$ is completely determined by that of $P$ and that a face of $E$ is exposed if and only if the corresponding face of $P$ is exposed. We apply these results to the convex hull of the image of a restricted momentum map.

math.CV↗