Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

Infinitesimal Automorphisms of VB-Groupoids and Algebroids

VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation VB-groupoid/algebroid.

Almost complex manifolds with non-degenerate torsion

We show that almost complex manifolds [Formula: see text] of real dimension 4 for which the image of the Nijenhuis tensor forms a non-integrable bundle, called torsion bundle, admit a [Formula: see text]-structure locally, that is, a double absolute parallelism. In this way, the problem of equivalence for such almost complex manifolds can be solved; moreover, the classification of locally homogeneous manifold [Formula: see text] is explicitly given when the Lie algebra of its infinitesimal automorphisms is non-solvable (indeed reductive). It is also shown that the group of the automorphisms of [Formula: see text] is a Lie group of dimension less than or equal to 4, whose isotropy subgroup has at most two elements, and that there are not non-constant holomorphic functions on [Formula: see text].

Regularity of infinitesimal CR automorphisms

We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if [Formula: see text] is such an infinitesimal CR automorphism, then [Formula: see text] is smooth for all multipliers [Formula: see text]. As an application, we study the regularity of infinitesimal automorphisms of certain infinite type hypersurfaces in [Formula: see text].

Essential Killing fields of parabolic geometries: projective and conformal structures

We use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.

The tangent complex and Hochschild cohomology of -rings

In this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

Let ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.