Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.     

Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring

A smooth stably complex manifold is said to be totally tangentially/normally split if its stably tangential/normal bundle is isomorphic to a sum of complex line bundles. It is proved that each class of degree greater than 2 in the graded unitary cobordism ring contains a quasitoric totally tangentially and normally split manifold.

     

A note on the {\rho}-Nitsche conjecture

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that if M is asymptotically harmonic of constant h = 0, then M is a flat manifold. This theorem shows that any asymptotically harmonic manifold in dimension 3 is a symmetric space, thus completing the classification of asymptotically harmonic manifolds in dimension 3.     

On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections

It is shown that if a closed smooth orientable manifold M ⁿ , n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.     

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kähler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP ^m or Gr2(C ^m+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.     

Finiteness of $$\pi _1$$-sensitive Hofer–Zehnder capacity and equivariant loop space homology

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer–Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.     

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E₁, F₁) ? · · · ? (Ek; Fk); where F _j := F |E _j , and each (E _j , F _j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).     

Torus Actions of Complexity 1 and Their Local Properties

We consider an effective action of a compact (n ? 1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n ? 1 has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold G_4,2, the complete flag manifold F₃, and quasitoric manifolds with an induced action of a subtorus of complexity 1.     

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = ?/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ? X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ? X which are characteristic complex submanifolds, i.e., embedding ? as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero(1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π : ? → X to minimal rational tangents of M.We prove that a compact characteristic complex submanifold S ? X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bnobtained by Mok(2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle TSas a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TXto S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ? X deduced from the results of Aubin(1978)and Yau(1978) which imply the existence of K¨ahler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension(depending on ?) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS,which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ? X for the case of the type-I domains of rank2 and the case of type-IV domains, and examine a case which is critical for both conjectures, i.e., on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in C~4.     

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.     

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian T -isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology H* _T (M; ?) with a point â in H ²(M;?) \ {0} up to weak H*(BT; ?)-algebra isomorphism preserving â.     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.     

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 \({\rm dim}_\mathbb{C} M\,\geq\,2\) .     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety Y is flexible if its special group SAut(Y) of automorphisms (generated by the elements of one-dimensional unipotent subgroups of Aut(Y)) acts transitively on Y, and an algebraic variety is stably flexible if its product with some affine space is flexible. An irreducible algebraic variety X is locally stably flexible if it is a union of a finite number of Zariski open sets each of which is stably flexible. The main result of this paper states that the blowup of a locally stably flexible variety along a smooth algebraic subvariety (not necessarily equidimensional or connected) is subelliptic, and, therefore, it is an Oka manifold.     

The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

We consider the classical N. Steenrod’s problem of realization of cycles by continuous images of manifolds. Our goal is to find a class \(\mathcal{M}_n \) of oriented n-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image of a manifold from the class \(\mathcal{M}_n \). We prove that as the class \(\mathcal{M}_n \) one can take a set of finite-fold coverings of the manifold M ⁿ of isospectral symmetric tridiagonal real (n + 1) × (n + 1) matrices. It is well known that the manifold M ⁿ is aspherical, its fundamental group is torsion-free, and its universal covering is diffeomorphic to ? ⁿ . Thus, every integral homology class of an arcwise connected space can be realized with some multiplicity by an image of an aspherical manifold with a torsion-free fundamental group. In particular, for any closed oriented manifold Q ⁿ , there exists an aspherical manifold that has torsion-free fundamental group and can be mapped onto Q ⁿ with nonzero degree.