Kahler Class Research Articles

Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality

On a compact Kahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kahler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.

Hermitian Yang–Mills Connections on Blowups

Consider a vector bundle over a Kahler manifold which admits a Hermitian Yang–Mills connection. We show that the pullback bundle on the blowup of the Kahler manifold at a collection of points also admits a Hermitian Yang–Mills connection, for Kahler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explicit. This recovers, through the Hitchin–Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.

Hirzebruch manifolds and positive holomorphic sectional curvature

This paper is the first step in a systematic project to study examples of Kahler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact Kahler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^1})$. We generalize Hitchin's construction and prove that any Hirzebruch manifold $M_{n, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^{n-1}})$ admits a Kahler metric of $H>0$ in each of its Kahler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kahler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all Kahler metrics of $H>0$ on a given Kahler manifold. We give higher dimensional examples such that some Kahler classes admit Kahler metrics with $H>0$ and some do not.

Volume minimization and conformally Kähler, Einstein–Maxwell geometry

Let $M$ be a compact complex manifold admitting a Kahler structure. A conformally Kahler, Einstein–Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde g$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^{2} \tilde g$ being a Kahler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kahler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kahler metric $g$ in the fixed Kahler class gives a cKEM metric $\tilde g = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kahler–Ricci solitons and Sasaki–Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kahler–Ricci soliton case and Sasaki–Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov–Maschler.

Metric contraction of the cone divisor by the conical Kähler–Ricci Flow

We use the momentum construction of Calabi to study the conical Kahler–Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov–Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts only the cone divisor to a single point and Gromov–Hausdorff converges to a two dimensional projective orbifold. The limiting behaviour depends only on the cone angle, numerical properties of the initial Kahler class, and the degree of the Hirzebruch surface. This gives the first example of the conical Kahler–Ricci flow contracting the cone divisor to a single point, and shows that the conical flow may contract curves of self-intersection less than $$(-\,1)$$ , as opposed to the smooth Kahler–Ricci flow. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kahler surfaces in general.

From ALE to ALF gravitational instantons

In this article, we give an analytic construction of ALF hyperkähler metrics on smooth deformations of the Kleinian singularity $\mathbb{C}^{2}/{\mathcal{D}}_{k}$, with ${\mathcal{D}}_{k}$ the binary dihedral group of order $4k$, $k\geqslant 2$. More precisely, we start from the ALE hyperkähler metrics constructed on these spaces by Kronheimer, and use analytic methods, e.g. resolution of a Monge–Ampère equation, to produce ALF hyperkähler metrics with the same associated Kähler classes.

Regularity of envelopes in Kähler classes

We establish the C^{1,1} regularity of quasi-psh envelopes in a Kahler class, confirming a conjecture of Berman.

Convergence of the $J$-flow on toric manifolds

We show that on a Kahler manifold whether the $J$-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the $J$-flow converges, verifying a conjecture in [19] (M. Lejmi and G. Szekelyhidi, “The $J$-flow and stability”) in this case. We also strengthen existing results on more general inverse $\sigma_k$ equations on Kahler manifolds.

The Kähler–Ricci flow on Fano bundles

We study the behavior of the Kahler–Ricci flow on some Fano bundles which is a trivial bundle on one Zariski open set. We show that if the fiber is $$\mathbb {P}^{m}$$ blown up at one point or some weighted projective space blown up at the orbifold point and the initial metric is in a suitable Kahler class, then the fibers collapse in finite time and the metrics converge sub-sequentially in Gromov–Hausdorff sense to a metric on the base.

A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds

In this study, we apply a result of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give some more homogeneous space involved proofs of recent classification of compact complex homogeneous locally conformal Kahler manifolds. In particular, we prove that the semisimple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semisimple Lie group S. We also prove that as an one dimensional complex torus bundle, the metrics on the manifold is completely determined by the metrics (which is the same as the Kahler class) on the base complex manifold and the metrics (same as the Kahler class) on the complex one dimensional torus.

Mass in Kähler Geometry

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kahler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kahler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kahler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kahler metrics, but also yields a Penrose-type inequality for the mass.

Anomalies, conformal manifolds, and spheres

The two-point function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal field theories (a.k.a. the conformal manifold). When the underlying quantum field theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N=(2,2) and N=(0,2) supersymmetric theories in d=2 and N=2 supersymmetric theories in d=4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is Kahler-Hodge and we further argue that it has vanishing Kahler class. For N=(2,2) theories in d=2 and N=2 theories in d=4 we also show that the relation between the sphere partition function and the Kahler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we find several examples of potential trace anomalies that obey the Wess-Zumino consistency conditions, but can be ruled out by a more detailed analysis.

Analysis and geometry of Floer theory of Landau-Ginzburg model on ℂ n

This article studies the Floer theory of Landau-Ginzburg (LG) model on Cn. We perturb the Kahler form within a fixed Kahler class to guarantee the transversal intersection of Lefschetz thimbles. The C0 estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.

Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds

We give a criterion for the coercivity of the Mabuchi functional for general Kahler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a Kahler-Einstein metric. We also prove the alpha invariant is a continuous function on the Kahler cone. As an application, we provide new Kahler classes on a general degree one del Pezzo surface for which the Mabuchi functional is coercive.

The consistency and convergence of K-energy minimizing movements

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing movements, these results imply that long time solutions to Calabi flow converge in the weak distance topology to minimizers of the K-energy functional on the metric completion of the space of Kahler metrics, assuming one exists.

Time-dependent Hermitian Yang–Mills flow on surfaces

We investigate a generalisation of Hermitian Yang–Mills flow in which the base metric itself is allowed to depend on the time-parameter. For technical reasons we restrict our attention to the case of a holomorphic vector bundles E over compact Riemann surfaces which is assumed to be slope stable with respect to a fixed Kahler class \(\tau \) on the base. We show that if \(\omega (t)\) is a family of Kahler metrics in \(\tau \) converging to a limit metric \(\omega _\infty \) at an exponential rate, then starting at a smooth initial Hermitian metric \(h_0\) on E, the Hermitian Yang–Mills flow with respect to the time-dependent metric \(\omega (t)\) admits a unique long-time solution h(t) converging exponentially to a \(\omega _\infty \)-Hermite–Einstein metric. The proof utilises an extension of Donadson’s techniques used to treat the case where the base metrics does not depend on the time-parameter.

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without $${\mathbb{C}}$$ -codimension-1 singularities. Let D be a suborbifold divisor in X such that $${D \supset {\rm Sing}(X)}$$ and −pK X = q[D] for some $${p, q \in \mathbb{N}}$$ with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler–Einstein, then, applying results from our previous paper (Conlon and Hein, Duke Math J, 162:2855–2902, 2013), we show that each Kahler class on $${X \setminus D}$$ contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r −1-e ) if X is smooth. This provides a definitive version of a theorem of Tian and Yau (Invent Math, 106:27–60, 1991). (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128)) when $${X = {\rm Bl}_p \mathbb{P}^{3}}$$ and $${D = {\rm Bl}_{p_1,p_2} \mathbb{P}^{2}}$$ is the strict transform of a smooth quadric through p in $${\mathbb{P}^3}$$ . Here D is no longer Kahler–Einstein, but the normal $${\mathbb{S}^1}$$ -bundle to D in X admits an irregular Sasaki–Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi–Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

Calabi flow, geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

We prove that constant scalar curvature Kahler metric �adjacent� to a fixed Kahler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the Calabi flow, from a new geometric perspective, and on the geometry of the space of Kahler metrics

A criterion for the properness of the $$K$$ K -energy in a general Kähler class

In this paper, we give a criterion for the properness of the \(K\)-energy in a general Kahler class of a compact Kahler manifold by using Song–Weinkove’s result in (Commun Pure Appl Math 61(2):210–229, 2008). As applications, we give some Kahler classes on \(\mathbb {C}\mathbb {P}^2\#3\overline{\mathbb {C}\mathbb {P}^2}\) and \(\mathbb {C}\mathbb {P}^2\#8\overline{\mathbb {C}\mathbb {P}^2}\) in which the \(K\)-energy is proper. Finally, we prove Song-Weinkove’s result on the existence of critical points of \(\hat{J}\) functional by the continuity method.

TheJ-flow on Kähler surfaces: a boundary case

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of negative self-intersection on the surface. We discuss an application to the Mabuchi energy functional on Kahler surfaces with ample canonical bundle.