Ahler Cone Research Articles

On the dual positive cones and the algebraicity of a compact K\"ahler manifold

We investigate the algebraicity of compact K\"ahler manifolds admitting a positive rational Hodge class of bidimension $(1,1)$. We prove that if the dual K\"ahler cone of a compact K\"ahler manifold $X$ contains a rational class as an interior point, then its Albanese variety is projective. As a consequence, we answer the Oguiso--Peternell problem for Ricci-flat compact K\"ahler manifolds. We also study related algebraicity problems for threefolds.

Calabi‐Yau metrics with cone singularities along intersecting complex lines: The unstable case

We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat polyhedral K\"ahler cone with conical singularities along two intersecting lines: one with cone angle corresponding to the line with smallest cone angle, while the other forms as the collision of the remaining lines into a single conical line. Using a branched covering argument, we can construct Calabi-Yau metrics with cone singularities along cuspidal curves with cone angle in the unstable range.

Swampland conjectures and infinite flop chains

We investigate swampland conjectures for quantum gravity in the context of M-theory compactified on Calabi-Yau threefolds which admit infinite sequences of flops. Naively, the moduli space of such compactifications contains paths of arbitrary geodesic length traversing an arbitrarily large number of K\"ahler cones, along which the low-energy spectrum remains virtually unchanged. In cases where the infinite chain of Calabi-Yau manifolds involves only a finite number of isomorphism classes, the moduli space has an infinite discrete symmetry which relates the isomorphic manifolds connected by flops. This is a remnant of the 11D Poincare symmetry and consequently gauged, as it has to be by the no-global symmetry conjecture. The apparent contradiction with the swampland distance conjecture is hence resolved after dividing by this discrete symmetry. If the flop sequence involves infinitely many non-isomorphic manifolds, this resolution is no longer available. However, such a situation cannot occur if the Kawamata-Morrison conjecture for Calabi-Yau threefolds is true. Conversely, the swampland distance conjecture, when applied to infinite flop chains, implies the Kawamata-Morrison conjecture under a plausible assumption on the diameter of the K\"ahler cones.

Contraction centers in families of hyperkähler manifolds

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperk\"ahler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the K\"ahler cone. Homology classes which can possibly be orthogonal to a wall of the K\"ahler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $b_2(M)>5$, this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has $b_2\leq 4$. Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.

Short Star-Products for Filtered Quantizations, I

We develop a theory of short star-products for filtered quantizations of graded Poisson algebras, introduced in 2016 by Beem, Peelaers and Rastelli for algebras of regular functions on hyperK\"ahler cones in the context of 3-dimensional $N=4$ superconformal field theories [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378]. This appears to be a new structure in representation theory, which is an algebraic incarnation of the non-holomorphic ${\rm SU}(2)$-symmetry of such cones. Using the technique of twisted traces on quantizations (an idea due to Kontsevich), we prove the conjecture by Beem, Peelaers and Rastelli that short star-products depend on finitely many parameters (under a natural nondegeneracy condition), and also construct these star products in a number of examples, confirming another conjecture by Beem, Peelaers and Rastelli.

Sasaki–Einstein metrics and K–stability

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.

Triality in little string theories

We study a class of eight-supercharge little string theories (LSTs) on the world-volume of $N$ M5-branes with transverse space $\mathbb{S}^{1}\times (\mathbb{C}^2/\mathbb{Z}_M)$. These M-brane configurations compactified on a circle are dual to $M$ D5-branes intersecting $N$ NS5-branes on $\mathbb{T}^2\times \mathbb{R}^{7,1}$ as well as to F-theory compactified on a toric Calabi-Yau threefold $X_{N,M}$. We argue that the K\"ahler cone of $X_{N,M}$ admits three regions associated with weakly coupled quiver gauge theories of gauge groups $[U(N)]^M, [U(M)]^N$ and $[U(\frac{NM}{k})]^k$ where $k=\mbox{gcd}(N,M)$. These provide low-energy descriptions of different LSTs. The duality between the first two gauge theories is well known and is a consequence of the S-duality between D5- and NS5-branes or the T-duality of the LSTs. The triality involving the third gauge theory is new and we demonstrate it using several examples. We also discuss implications of this triality for the W-algebras associated with the Alday-Gaiotto-Tachikawa dual theories.

Kählerity of Shrinking Gradient Ricci Solitons Asymptotic to Kähler Cones

We prove that a shrinking gradient Ricci soliton which is asymptotic to a K\"ahler cone along some end is itself K\"ahler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally K\"ahler.

Toric Vaisman manifolds

Vaisman manifolds are strongly related to K\"ahler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if a complete Vaisman manifold is toric, then the associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki manifold, whose K\"ahler cone is equipped with an appropriate compatible action, gives rise to a toric Vaisman manifold. In the special case of a strongly regular compact Vaisman manifold, we show that it is toric if and only if the corresponding K\"ahler quotient is toric.

Geodesics in the space of Kähler cone metrics, I

In this paper, we study the Dirichlet problem of the geodesic equation in the space of K\"ahler cone metrics ${\cal H}_{\cal B}$; that is equivalent to a homogeneous complex Monge-Amp\`ere equation whose boundary values consist of K\"ahler metrics with cone singularities. Our approach concerns the generalization of the function space defined by Donaldson to the case of K\"ahler manifolds with boundary; moreover we introduce a subspace ${\cal H}_C$ of ${\cal H}_{\cal B}$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_{\cal B}$ geodesics whose boundary values lie in ${\cal H}_C$. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2, \alpha}_{\cal B}$ approximate geodesics under the $C^{1,1}_{\cal B}$-norm. As a geometric application, we prove the metric space structure of ${\cal H}_C$.

Infinite number of MSSMs from heterotic line bundles?

We consider heterotic ${\mathrm{E}}_{8}\ifmmode\times\else\texttimes\fi{}{\mathrm{E}}_{8}$ supergravity compactified on smooth Calabi-Yau manifolds with line bundle gauge backgrounds. Infinite sets of models that satisfy the Bianchi identities and flux quantization conditions can be constructed by letting their background flux quanta grow without bound. Even though we do not have a general proof, we find that all examples are at the boundary of the theory's validity: the Donaldson-Uhlenbeck-Yau equations, which can be thought of as vanishing D-term conditions, cannot be satisfied inside the K\"ahler cone unless a growing number of scalar vacuum expectation values is switched on. As they are charged under various line bundles simultaneously, the gauge background gets deformed by these VEVs to a non-Abelian bundle. In general, our physical expectation is that such infinite sets of models should be impossible, since they never seem to occur in exact conformal field theory constructions.

Rotational symmetry of conical Kähler–Ricci solitons

We show that expanding K\"ahler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to K\"ahler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

Momentum construction on Ricci-flat Kähler cones

We extend Calabi ansatz over K\"ahler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat K\"ahler metric on K\"ahler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat K\"ahler metric on the toric K\"ahler cone manifold constructed from a toric diagram with a constant height.

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution of a Ricci-flat K\"ahler cone X admits a complete Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\"ahler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat K\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.

Gauged supergravity and singular Calabi-Yau manifolds

Compactifications of M-theory on singular manifolds contain additional charged massless states descending from M-branes wrapped on vanishing cycles. We construct the first explicit example of a complete supergravity Lagrangian that includes such extra states. This is done for a compactification on a Calabi-Yau threefold that develops a genus zero curve of A_1 singularities at the boundary of the K\"ahler cone with a resulting SU(2) gauge symmetry enhancement. The corresponding SU(2) gauged supergravity Lagrangian includes two charged and two neutral vector multiplets, and turns out to be uniquely fixed by the Calabi-Yau geometry and by the effective ungauged Lagrangian describing the Coulomb branch. One can see explicitly how resolving the singularity corresponds to a supersymmetric Higgs effect in the gauged supergravity description. The elementary transformation relating the two families of smooth Calabi-Yau resolutions of the singularity acts as the SU(2) Weyl twist. The resulting structure appears to be very rigid and is likely to apply to other types of singularities and manifolds as well.

Constructing Kähler-Ricci Solitons from Sasaki-Einstein Manifolds

We construct gradient K\"ahler-Ricci solitons on Ricci-flat K\"ahler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the results generalize constructions of Cao and Feldman-Ilmanen-Knopf.