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Gabriella Clemente

Publications and source records attributed to Gabriella Clemente.

13 recordsLinked to original sources
arXiv2026

K\"ahler thresholds

The topology of K\"ahler manifolds is largely determined by the geometry due to its rigidity. In particular, the cohomology and Hodge theory of compact K\"ahler manifolds is quite restricted. We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate K\"ahler condition are even, and the even Betti numbers are strictly positive. We call this new type of degenerated K\"ahler manifold acK. Our approach to proving these results makes use of a compactness theorem for acK manifolds, and a new version of Hodge theory for compact manifolds endowed with a Sobolev regular K\"ahler strucutre. In addition, we lay out a program to pursue the study of acK geometry that accommodates not only the classical viewpoint, but also constructive and finitary proofs, as well as formalization with proof assistants.

math.AG↗
arXiv2026

Synthetic Differential Geometry in Lean

This article is about the formalization of synthetic differential geometry with the Lean proof assistant and the mathematical library mathlib. The main result we prove and formalize is a Taylor theorem for functions of several variables, where the series expansion is around an infinitesimal neighborhood. Most of our proofs are in fact new. Our investigations highlight the possibility of using mathlib to do constructive mathematics.

cs.LO↗
arXiv2025

Real Einstein loci

The aim of this article is to study the interplay between the complex, and underlying real geometries of a K\"ahler manifold. We provide a necessary and sufficient condition for certain anti-holomorphic automorphisms of a compact K\"ahler-Einstein manifold to determine real Einstein submanifolds.

math.DG↗
arXiv2023

Curvature, integrability, and the six sphere

This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not hermitian. The partial proof is intrinsic in nature, and shows some level of promise when it comes to generalizing the non-complexity of the round $S^6$ result in new directions.

math.DG↗
arXiv2023

A curvature obstruction to integrability

The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of Riemannian metrics can obstruct the integrability of almost-complex structures. It is shown that certain special complex structures cannot coexist with non-flat constant curvature metrics, and a formal variational realization of these structures is provided. The approach followed here is direct, meaning that it bypasses the classical theory. The idea is to find obstruction equations for the integrability of almost-complex structures by way of Nijenhuis tensor derivatives. These new equations involve the curvature of a torsion-free connection, and reveal the interplay between the almost-complex and Riemannian geometries. Curvature scalars to detect non-complexity in the compact case then arise in a natural way.

math.DG↗
arXiv2022

Geometry of universal embedding spaces for almost complex manifolds

We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the "transverse embeddability" of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of the integrability locus characterizing integrable almost complex structures. Our approach can potentially be used to investigate the existence (or non-existence) of topological obstructions to integrability.

math.AG↗
arXiv2022

Geometric structures as variational objects, II

This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.

math.DG↗
arXiv2022

Complex structures as critical points

This note aims at obtaining a variational characterization of complex structures by means of a calculus of variations for real vector bundle valued differential forms, and outlines a perspective to study existence questions via functionals and stability notions.

math.DG↗
arXiv2014

Graphs with minimal well-covered dimension

There is a class of graphs with well-covered dimension equal to the simplicial clique number that contains all chordal graphs and infinitely many other graphs. These graphs generalize a result by Brown and Nowakowski on the well-covered dimension of chordal graphs. Furthermore, each member of the infinite family of Sierpinski gasket graphs of order at least $2$ has well-covered dimension $3,$ the simplicial clique number.

math.CO↗