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Lorenzo Sillari

Publications and source records attributed to Lorenzo Sillari.

13 recordsLinked to original sources
arXiv2026

Invariant forms compute the Dolbeault cohomology of complex nilmanifolds

We prove that the inclusion of left-invariant forms into the Dolbeault complex of a compact nilmanifold $M$ endowed with a left-invariant complex structure $J$ induces an isomorphism in cohomology in every bidegree, settling a long-standing conjecture. As consequences, we show that small deformations of $J$ are still invariant, as conjectured by Hasegawa. We also prove that Bott--Chern, Aeppli, and Frölicher invariants are computed by invariant forms and are independent of the lattice, settling a conjecture of Angella on Bott--Chern cohomology.

math.DG↗
arXiv2026

Classification of homogeneous almost complex $4$-manifolds with non-degenerate torsion bundle

We investigate the local and global geometry of almost complex $4$-manifolds admitting non-degenerate torsion bundle. The rigidity of these structures forces a parallelizable $J$-adapted double cover, which imposes severe topological constraints on the underlying manifold. Exploiting this rigidity, we give a complete classification in the homogeneous setting. We show that such a manifold is diffeomorphic either to a $4$-dimensional Lie group carrying an almost complex structure with non-degenerate torsion bundle, or to a product $L(4,1)\times\mathbb R$ or $L(4,1)\times\mathbb T$, where $L(4,1)$ is a lens space. We also determine exactly which real $4$-dimensional Lie algebras admit such a structure. Constructively, we realize every admissible algebra by an explicit invariant structure, thereby closing the existence question in dimension $4$. We also relate these structures to certain Engel structures that we call Nijenhuis--Engel, and answer the resulting existence questions in the homogeneous case.

math.DG↗
arXiv2025

Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class

In this paper we investigate the Kodaira dimension of almost complex $4$-manifolds with torsion first Chern class. First, we prove that, if the almost complex structure is also tamed, the only possible values for the Kodaira dimension are $0$ or $-\infty$. This is done by developing the theory of pseudoholomorphic structures on vector bundles. In arbitrary dimension, we study infinitesimal deformations of structures with pseudoholomorphically torsion canonical bundle. We compute their tangent space and, under suitable assumptions, we prove an unobstructedness theorem in the spirit of Bogomolov--Tian--Todorov. Together, our results allow to fully describe non-integrable infinitesimal deformations of complex structures on $K3$ and Enriques surfaces in terms of their Kodaira dimension.

math.DG↗
arXiv2025

Kodaira dimension of $\mathrm{SU}(m)$-structures

We study the Kodaira dimension of almost complex manifolds admitting an $\mathrm{SU} (m)$-structure. We introduce the notion of almost complex structure of splitting type and of associated $\mathrm{SU} (m)$-structure. When the latter is pseudoholomorphic, we provide two constructions that allow to obtain non-invariant almost complex structures with Kodaira dimension $0$, resp.\ with Kodaira dimension $-\infty$. Our results apply, in particular, to complex structures of splitting type and to several almost complex manifolds already well-studied in the literature

math.DG↗
arXiv2025

Invariant and non-invariant almost complex structures on compact quotients of Lie groups

In this paper we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of invariant and non-invariant structures, with a special attention to their canonical bundle and Kodaira dimension. We provide new examples of computations of Kodaira dimension of invariant and non-invariant structures.

math.DG↗
arXiv2024

Real analytic curves of almost complex structures

We prove that, on a compact almost complex manifold, the space of almost complex structures whose Nijenhuis tensor has rank at least $k$ at every point is either empty or dense in each path-connected component of the space of almost complex structures. In particular, this applies to maximally non-integrable almost complex structures.

math.DG↗
arXiv2023

On the spaces of $(d+d^c)$-harmonic forms and $(d+d^Λ)$-harmonic forms on almost Hermitian manifolds and complex surfaces

We study the spaces of $(d + d^c)$-harmonic forms and $(d + d^Λ)$-harmonic forms, the natural generalization of the spaces of Bott-Chern harmonic forms, resp. symplectic harmonic forms from complex, resp. symplectic, manifolds to almost Hermitian manifolds. With the same techniques, we also prove that Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold, a fact that was well-known for Hodge numbers of compact complex surfaces. We give several applications to compact quotients of Lie groups by a lattice.

math.DG↗
arXiv2023

On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott-Chern and Aeppli cohomologies defined using the operators $d$, $d^c$. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to $d$, $d^c$, showing their relation with Bott-Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott-Chern cohomology of $1$-forms is finite-dimensional on compact manifolds and provides an almost complex invariant $h^1_{d + d^c}$ that distinguishes between almost complex structures. On almost Kähler $4$-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

math.DG↗
arXiv2022

Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on $6$-nilmanifolds and the associated Lie algebras. We classify which $6$-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for left-invariant structures on $6$-nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for left-invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.

math.DG↗
arXiv2022

Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along $\mathcal{J}$-symplectic submanifolds. We also export branched coverings to the generalized complex setting. As an application, stable generalized complex structures are produced on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.

math.DG↗
arXiv2021

On the minimal number of solutions of the equation $ ϕ(n+k)= M \, ϕ(n) $, $ M=1$, $2$

We fix a positive integer $k$ and look for solutions of the equations $ϕ(n+k) = ϕ(n)$ and $ϕ(n + k) = 2ϕ(n)$. We prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even and five for the second one when $k$ is odd. These results hold for $k \le 2 \cdot 10^{100}$. We also show that for the second equation with even $k$ there are at least three solutions for $k \le 4 \cdot 10^{58}$. Our work increases the previous minimal number of known solutions for both equations.

math.NT↗
arXiv2020

Dolbeault and $J$-invariant cohomologies on almost complex manifolds

In this paper we relate the cohomology of $J$-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S.O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.

math.DG↗