Ahler Manifolds Research Articles

Reduction of manifolds with semi-negative holomorphic sectional curvature

In this note, we continue the investigation of a projective K\"ahler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of $M$. We also prove a splitting theorem for $M$ in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.

Compact complex manifolds with small Gauduchon cone

This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact K\"ahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main idea is to explore the link between the {\it class} ${\cal C}$ property and the closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a fact leading to the study of semi-continuity properties under deformations of the complex structure of the dual cones of cohomology classes of such currents and of Gauduchon metrics. Our main finding is a new class of compact complex, possibly non-K\"ahler, manifolds defined by the condition that every Gauduchon metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be small in a certain sense. We term them sGG manifolds and find numerical characterisations of them in terms of certain relations between various cohomology theories (De Rham, Dolbeault, Bott-Chern, Aeppli). We also produce several concrete examples of nilmanifolds demonstrating the differences between the sGG class and well-established classes of complex manifolds. We conclude that sGG manifolds enjoy good stability properties under deformations and modifications.

Canonical almost geodesic mappings of type $\underset\theta\pi{}_2(0,F)$, ${\small\theta\in\{1,2\}}$ between generalized parabolic K\"ahler manifolds

Canonical almost geodesic mappings of type $\underset\theta\pi{}_2(0,F)$, ${\small\theta\in\{1,2\}}$ between generalized parabolic K\"ahler manifolds

ASK/PSK-correspondence and the r-map

We formulate a correspondence between affine and projective special K\"ahler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special K\"ahler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special K\"ahler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative $\alpha'$-corrections in heterotic and type-II string compactifications with $N=2$ supersymmetry. Also affine special K\"ahler manifolds with quadratic prepotential are mapped to one-parameter families of projective special K\"ahler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.

Slope instability of projective spaces blown up along a line

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a K\"ahler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature K\"ahler metrics in any rational K\"ahler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in K\"ahler classes that are close to the pullback of the Fubini--Study class.

Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.

Convexity of the extended K-energy and the large time behavior of the weak Calabi flow

Let $(X,\omega)$ be a compact connected K\"ahler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of K\"ahler potentials $\mathcal H_\omega$ with respect to the $L^p$-type path length metric $d_p$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\mathcal E^p$ is a $d_p$-lsc functional that is convex along finite energy geodesics. Second, following the program of J. Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space $(\mathcal E^2,d_2)$. This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the $d_2$-metric or it $d_1$-converges to some minimizer of the K-energy inside $\mathcal E^2$. This gives the first concrete result about the long time convergence of this flow on general K\"ahler manifolds, partially confirming a conjecture of Donaldson. Finally, we investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is K\"ahler. If the twisting form is only smooth, we reduce this problem to a conjecture on the regularity of minimizers of the K-energy on $\mathcal E^1$, known to hold in case of Fano manifolds.

Asymptotic properties of extremal Kähler metrics of Poincaré type

Consider a compact K\"ahler manifold X with a simple normal crossing divisor D, and define Poincar\'e type metrics on X\D as K\"ahler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincar\'e type K\"ahler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) K\"ahler metric, possibly of Poincar\'e type, on every component of D. We also show that when the divisor is smooth, the constant scalar curvature/extremal metric on X\D is asymptotically a product near the divisor.

Open Gromov–Witten invariants, mirror maps, and Seidel representations for toric manifolds

Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.

Kähler Manifolds with Negative Holomorphic Sectional Curvature, Kähler-Ricci Flow Approach

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact K\"ahler manifolds. In this short note, we give anothe proof of their theorems by using the K\"ahler-Ricci flow.

On the tangent cone of Kähler manifolds with Ricci curvature lower bound

Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete K\"ahler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the K\"ahler case. We also discuss some applications to complete K\"ahler manifolds with nonnegative bisectional curvature.

Geometry of special Galileons

A theory known as special Galileon has recently attracted considerable interest due to its peculiar properties. It has been shown that it represents an extremal member of the set of effective field theories with an enhanced soft limit. This property makes its tree-level $S$-matrix fully on-shell reconstructible and representable by means of the Cachazo-He-Yuan representation. The enhanced soft limit is a consequence of new hidden symmetry of the special Galileon action; however, until now, the origin of this peculiar symmetry has remained unclear. In this paper we interpret this symmetry as a special transformation of the coset space $\mathrm{GAL}(D,1)/SO(1,D\ensuremath{-}1)$ and show that there exists a three-parametric family of invariant Galileon actions. The latter family is closed under the duality which appears as a natural generalization of the above mentioned symmetry. We also present a geometric construction of the special Galileon action using a $D$-dimensional brane propagating in $2D$-dimensional flat pseudo-Riemannian space. Within such a framework, the special Galileon symmetry emerges as a $U(1,D\ensuremath{-}1)$ symmetry of the target space, which can be treated as a $D$-dimensional K\"ahler manifold. Such a treatment allows for classification of the higher order invariant Lagrangians needed as counterterms on the quantum level. We also briefly comment on the relation between such higher order Lagrangians and the Lagrangians that are invariant with respect to the polynomial shift symmetry.

Varieties with $\mathbb {P}$-units

We study the class of compact Kahler manifolds with trivial canonical bundle and the property that the cohomology of the trivial line bundle is generated by one element. If the square of the generator is zero, we get the class of strict Calabi--Yau manifolds. If the generator is of degree 2, we get the class of compact hyperkahler manifolds. We provide some examples and structure results for the cases where the generator is of higher nilpotency index and degree. In particular, we show that varieties of this type are closely related to higher-dimensional Enriques varieties.

Projectively related metrics, Weyl nullity and metric projectively invariant equations

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries which is defined on a rank $(n+1)$-bundle. We show this connection is linked to the metrisability equations that govern the existence of metrics compatible with the structure. The fundamental 2-tensor also leads to a new class of invariant linear differential operators that are canonically associated to these geometries; included is a third equation studied by Gallot et al. We apply the results to study the metrisability equation, in the nullity setting described. We obtain strong local and global results on the nature of solutions and also on the nature of the geometries admitting such solutions, obtaining classification results in some cases. We show that closed Sasakian and K\"ahler manifold do not admit nontrivial solutions. We also prove that, on a closed manifold, two nontrivially projectively equivalent metrics cannot have the same tracefree Ricci tensor. We show that on a closed manifold a metric having a nontrivial solution of the metrisablity equation cannot have two-dimensional nullity space at every point. In these statements the meaning of trivial solution is dependent on the context. There is a function $B$ naturally appearing if a metric projective structure has nullity. We analyse in detail the case when this is not a constant, and describe all nontrivially projectively equivalent Riemannian metrics on closed manifolds with nonconstant $B$.

A Tannakian approach to dimensional reduction of principal bundles

Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact K\"ahler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduction of holomorphic principal bundles over $X\times G/P$. The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of $G$-equivariant principal bundles over $X\times G/P$, and to establish a Hitchin--Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that $X$ is a complex projective manifold.

Dynamical Crossing of an Infinitely Degenerate Critical Point

We study the evolution of a driven harmonic oscillator with a time-dependent frequency $\omega_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past then the asymptotic future state is a squeezed state whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points where infinitely many eigenvalues collide. We extend our analysis to the situation where the gap at $t=0$ remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincar\'e disk model viewed as a K\"ahler manifold.

Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

We show that any dimension $6$ nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para\nobreakdash-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry, this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para\nobreakdash-)K\"ahler manifolds, including how the almost (para\nobreakdash-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.

Versions of injectivity and extension theorems

We give an analytic version of the injectivity theorem by using multiplier ideal sheaves, and prove some extension theorems for the adjoint bundle of dlt pairs. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry and the Strominger-Yau-Zaslow conjecture for hyperKahler manifolds.

Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.

On holomorphically projective mappings of parabolic K\"ahler manifolds

On holomorphically projective mappings of parabolic K\"ahler manifolds