arXiv research

Classifies hyperbolic manifolds with specific automorphism groups.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by …

We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel s…

Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.

We study the group of leafwise holomorphic smooth automorphisms of Reeb components of leafwise complex foliation which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we determine the structure of the group of leafwise holomorphi…

Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

The study of real Einstein submanifolds in Kähler geometry.

Study on automorphisms of complex $b^k$-manifolds, extending previous work.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.

Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.

The paper constructs contact-hyperbolic manifolds with large automorphism groups.

Study of Hermitian structures on toric suspensions of balanced manifolds.

In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexification $\mathcal{M}$ of a generic real analytic submanifold $M \subseteq \C^N$ of finite type at some point $p$ into the complexification $\mathcal{M}'$ of a generic real analytic s…

We obtain a complete classification of complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$, for which the dimension of the group of holomorphic automorphisms is equal to $n^2$.

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the gro…

We study local automorphisms of holomorphic Cartan geometries. This leads to classification results for compact complex manifolds admitting Cartan geometries. We prove that a compact Calabi-Yau manifold bearing a holomorphic Cartan geometry of algebraic type admits a finite unramified cover which is a complex torus.

Unique optimal symplectic connections found for submersions.

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 4$ whose group of holomorphic automorphisms has dimension either $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series of articles that achieve classifications for automorphism group dimension $n^2-3$ and greater.

In this paper we determine all Kobayashi-hyperbolic 2-dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of hyperbolic $n$-dimensional manifolds, with automorphism group of dimension at least $…

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.

Study of flows on complex manifolds with holomorphic properties.

Logarithmic connections on complex manifolds with trivial tangent bundle.

We consider complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ for which the dimension of the group of holomorphic automorphisms is equal to $n^2-1$. We give a complete classification of such manifolds for $n\ge 3$ and discuss several examples for $n=2$.

Study proper holomorphic maps between specific domains, proving rigidity under certain conditions.

New examples of translation surfaces on hyperelliptic curves with many automorphisms.

Compact pseudo-Hermitian spaces have rigid holomorphic isometries.

A connected Fano complex-contact manifold is isomorphic to the kaehlerian C-space of Boothby type with a natural complex-contact structure corresponding to a non-abelian simple complex Lie algebra if the contact line bundle is very ample. A. Beauville relaxed the provision to two assumptions that the contact line bundl…

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of…

Given an l-component pointed oriented link (L,p) in an oriented three-manifold Y, one can construct its link Floer chain complex CFL(Y,L,p) over the polynomial ring F_2[U_1,...,U_l]. Moving the basepoint p_i in the link component L_i once around induces an automorphism of CFL(Y,L,p). In this paper, we study an automorp…

A manifold $M$ is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. Let $M$ be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on its covering. We show that $M$ admits an automorphic pote…

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the…

Holomorphic actions on complex spaces for nilpotent groups.

Study on opers over complex manifolds of dimension one.

Develops Lie algebraic approach for compact complex homogeneous manifolds.

We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra co…

The equivariant holomorphic torsion of a compact locally symmetric manifold and an automorphism is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…

We review the standard Hopf construction of Reeb components with leafwise complex structure and determine the group of leafwise holomorphic smooth automorphisms for tame Reeb components in the case of complex leaf dimension one. For this, we solve the Schröder type functional equation on the half line for expanding dif…

Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse holomorphic structure is trivial.

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…

The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components ob…

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivari…

In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphis…

Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.