Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kähler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kähler angle if it is neither holomorphic nor anti-holomorphic with Kähler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).     

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.     

Calabi-Yau manifolds with isolated conical singularities

Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\) such that \(-\frac{1}{q}K_{Z}\) is very ample and such that the germ \((X,x)\) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of \(\frac{1}{q}K_{Z}\). We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

The deformation of symplectic critical surfaces in a Kähler surface-II—compactness

In this paper we consider the compactness of \(\beta \)-symplectic critical surfaces in a Kähler surface. Let M be a compact Kähler surface and \(\Sigma _i\subset M\) be a sequence of closed \(\beta _i\)-symplectic critical surfaces with \(\beta _i\rightarrow \beta _0\in (0,\infty )\). Suppose the quantity \(\int _{\Sigma _i}\frac{1}{\cos ^q\alpha _i}d\mu _i\) (for some \(q>4\)) and the genus of \(\Sigma _{i}\) are bounded, then there exists a finite set of points \({{\mathcal {S}}}\subset M\) and a subsequence \(\Sigma _{i'}\) which converges uniformly in the \(C^l\) topology (for any \(l<\infty \)) on compact subsets of \(M\backslash {{\mathcal {S}}}\) to a \(\beta _0\)-symplectic critical surface \(\Sigma \subset M\), each connected component of \(\Sigma \setminus {{\mathcal {S}}}\) can be extended smoothly across \({{\mathcal {S}}}\).     

Isospectral nearly Kähler manifolds

We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\). We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.     

Slope instability of projective spaces blown up along a line

Let \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) be a Kähler manifold obtained by blowing up a complex projective space \(\mathbb {P}^n\) along a line \(\mathbb {P}^1\). We prove that \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kähler metrics in any rational Kähler class.     

Gaussian Analytic Functions in the Polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of \(\mathbb {C}^n\). Following the scheme previously used in the unit ball, we first study the asymptotics of fluctuations of linear statistics as the directional intensities \(L_j\), \(j=1,\dots ,n\) tend to \(\infty \). Then, we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact Kähler manifolds.     

Equitorsion holomorphically projective mappings of generalized Kählerian space of the first kind

In this paper we define generalized Kählerian spaces of the first kind (\(G_1^K N\)) given by (2.1)–(2.3). For them we consider hollomorphically projective mappings with invariant complex structure. Also, we consider equitorsion geodesic mapping between these two spaces (\(G_1^K N\) and \(G_1^{\bar K} N\)) and for them we find invariant geometric objects.     

Proper Modifications of Generalized <Emphasis Type="Italic">p</Emphasis>-Kähler Manifolds

In this paper, we consider a proper modification \(f : \tilde{M} \rightarrow M\) between complex manifolds, and study when a generalized p-Kähler property goes back from M to \(\tilde{M}\). When f is the blow-up at a point, every generalized p-Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for \(p=1\). For \(p=n-1\), we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kähler manifolds.     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

Complex geometry of moment-angle manifolds

Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold \(\mathcal {Z}\) is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold \(\mathcal {Z}\) is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold \(\mathcal {Z}\) is equipped with a canonical holomorphic foliation \({\mathcal {F}}\) which is equivariant with respect to the \(({\mathbb {C}}^\times )^m\)-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi–Eckmann manifolds, and their deformations. We construct transversely Kähler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kähler submanifold (or, more generally, a Fujiki class \(\mathcal {C}\) subvariety) in such a moment-angle manifold is contained in a leaf of the foliation \({\mathcal {F}}\). For a generic moment-angle manifold \(\mathcal {Z}\) in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of \(\mathcal {Z}\) is zero.     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.     

Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures

Let (M, g, J) be a compact Hermitian manifold and \(\Omega\) the fundamental 2-form of (g, J). A Hermitian manifold (M, g, J) is called a locally conformal Kähler manifold if there exists a closed 1-form α such that \(d\Omega=\alpha \wedge \Omega\) . The purpose of this paper is to give a completely classification of locally conformal Kähler nilmanifolds with left-invariant complex structures.     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

General Natural $$(\alpha ,\varepsilon )$$-Structures

We study in a unified way the \((\alpha ,\varepsilon )\)-structures of general natural lift type on the tangent bundle of a Riemannian manifold. We characterize the general natural \(\alpha \)-structures on the total space of the tangent bundle of a Riemannian manifold, and provide their integrability conditions (the base manifold is a space form and some involved coefficients are rational functions of the other ones). Then, we characterize the two classes (with respect to the sign of \(\alpha \varepsilon \)) of \((\alpha ,\varepsilon )\)-structures of general natural type on TM. The class \(\alpha \varepsilon =-1\) is characterized by some proportionality relations between the coefficients of the metric and those of the \(\alpha \)-structure, and in this case, the structure is almost Kählerian if and only if the first proportionality factor is the derivative of the second one. Moreover, the total space of the tangent bundle is a Kähler manifold if and only if it depends on three coefficients only (two coefficients of the integrable \(\alpha \)-structure and a proportionality factor).     

Complete real Kähler submanifolds in codimension two

Minimal isometric immersions \(f : M^{2n} \rightarrow {\mathbb{R}}^{2n+2}\) in codimension two from a complete Kähler manifold into Euclidean space had been classified in Dajczer and Gromoll (Invent Math 119:235–242, 1995) for n ≥  3. In this note we describe the non-minimal situation showing that, if f is real analytic but not everywhere minimal, then f is a cylinder over a real Kähler surface \(g : N^4 \rightarrow {\mathbb{R}}^6\) , that is, \(M^{2n} = N^4 \times {\mathbb{C}}^{n-2}\) and f = g × id split, where \({id} : {\mathbb{C}}^{n-2} \cong {\mathbb{R}}^{2n-4}\) is the identity map. Moreover, g can be further described.     

Double ramification cycles on the moduli spaces of curves

Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle \(\mathsf{DR}_{g}(\mu,\nu)\) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for \(\mathsf{DR}_{g}(\mu,\nu)\) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.When \(\mu=\nu=\emptyset\), the formula for double ramification cycles expresses the top Chern class \(\lambda_{g}\) of the Hodge bundle of \(\overline {\mathcal{M}}_{g}\) as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.     

A Higgs bundle on a Hermitian symmetric space

Let \(M := \Gamma\backslash G/K\) be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice \(\Gamma\subset G\) . There is a natural flat principal G-bundle over the compact Kähler manifold M which is constructed from the principal Γ-bundle over M defined by the quotient map \(G/K\longrightarrow M\) . We construct the principal G-Higgs bundle over M corresponding to this flat G-bundle. This principal G-Higgs bundle is rigid if \({\rm dim}_\mathbb{C} M\,\geq\,2\) .     

Kähler metrics with cone singularities along a divisor of bounded Ricci curvature

Let D be a smooth divisor in a compact complex manifold X and let \(\beta \in (0,1)\). We use the liner theory developed by Donaldson (Essays in Mathematics and Its Applications, Springer, Berlin, pp 49–79, 2012) to show that in any positive co-homology class on X there is a Kähler metric with cone angle \(2\pi \beta \) along D which has bounded Ricci curvature. We use this result together with the Aubin–Yau continuity method to give an alternative proof of a well-known existence theorem for Kähler–Einstein metrics with cone singularities.