A counterexample to gluing theorems for MCP metric measure spaces

Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature $\geq \kappa$ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the $\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane, which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.Comment: 10 pages, 2 figures. Accepted version, to appear on the Bulletin of the London Mathematical Societ

Measure contraction properties of Carnot groups

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number $k+3$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least $N$ such that the $\mathrm{MCP}(0,N)$ is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.Comment: 17 pages, final version, to appear on "Calculus of Variations and PDEs

Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of $3$-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete $3$-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\pi$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations: \begin{equation*} \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \to \mathbb{HP}^d, \end{equation*} whose exact sub-Riemannian diameter is $\pi$, for all $d \geq 1$.Comment: 34 pages, v2: fixed and clarified the proof of Theorem 7 and some typos, v3: final version, to appear on Journal of the Institute of Mathematics of Jussie

Generalized adjoint forms on algebraic varieties

We prove a full generalization of the Castelnuovo's free pencil trick. We show its analogies with the Adjoint Theorem; see L. Rizzi, F. Zucconi, Differential forms and quadrics of the canonical image, arXiv:1409.1826 and also Theorem 1.5.1 in G. P. Pirola, F. Zucconi, Variations of the Albanese morphisms, J. Algebraic Geom. 12 (2003), no. 3, 535-572. Moreover we find a new formulation of the Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic $1$-forms.Comment: 18 page

A note on Torelli-type theorems for Gorenstein curves

Using the notion of generalized divisors introduced by Hartshorne, we adapt the theory of adjoint forms to the case of Gorenstein curves. We show an infinitesimal Torelli-type theorem for vector bundles on Gorenstein curves. We also construct explicit counterexamples to the infinitesimal Torelli claim in the case of a reduced reducible Gorenstein curve.Comment: 17 page

Differential forms and quadrics of the canonical image

Let $\pi\colon\mathcal{X}\to B$ be a family over a smooth connected analytic variety $B$, not necessarily compact, whose general fiber $X$ is smooth of dimension $n$, with irregularity $\geq n+1$ and such that the image of the canonical map of $X$ is not contained in any quadric of rank $\leq 2n+3$. We prove that if the Albanese map of $X$ is of degree $1$ onto its image then the fibers of $\pi\colon\mathcal{X}\to B$ are birational under the assumption that all the $1$-forms and all the $n$-forms of a fiber are holomorphically liftable to $\mathcal{X}$. Moreover we show that generic Torelli holds for such a family $\pi\colon \mathcal{X}\to B$ if, in addition to the above hypothesis, we assume that the fibers are minimal and their minimal model is unique. There are counterexamples to the above statements if the canonical image is contained inside quadrics of rank $\leq 2n+3$. We also solve the infinitesimal Torelli problem for an $n$-dimensional variety $X$ of general type with irregularity $\geq n+1$ and such that its cotangent sheaf is generated and the canonical map is a rational map whose image is not contained in a quadric of rank less or equal to $2n+3$.Comment: 23 pages, revised version incorporating referees' comments, exposition improve

On Jacobi fields and canonical connection in sub-Riemannian geometry

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and we discuss some of its properties.Comment: 13 pages, (v2) minor corrections. Final version to appear on Archivum Mathematicu

Comparison theorems for conjugate points in sub-Riemannian geometry

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet-Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.Comment: 33 pages, 5 figures, v2: minor revision, v3: minor revision, v4: minor revisions after publicatio