On Grauert–Riemenschneider Type Criteria

Let(X, ω) be a compact Hermitian manifold of complex dimension n. In this article,we first survey recent progress towards Grauert–Riemenschneider type criteria. Secondly, we give a simplified proof of Boucksom's conjecture given by the author under the assumption that the Hermitian metric ω satisfies ?■ω~l= for all l, i.e., if T is a closed positive current on X such that ∫_XT_(ac)~n 0, then the class {T } is big and X is Kahler. Finally, as an easy observation, we point out that Nguyen's result can be generalized as follows: if ?■ω = 0, and T is a closed positive current with analytic singularities,such that ∫_XT_(ac)~n 0, then the class {T} is big and X is Kahler.     

Simple root flows for Hitchin representations

On a Kähler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kähler condition. While such a link is not so obvious in the non-Kähler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the Kähler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kähler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Sur la géométrie systolique des variétés de Bieberbach

We review the theory of quaternionic Kähler and hyperkähler structures. Then we consider the tangent bundle of a Riemannian manifold M endowed with a metric connection D, with torsion, and with its well estabilished canonical complex structure. With an almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM, which is quaternionic Kähler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM, in the wider context of geometry with torsion. Finally we discover an S ³-bundle of complex structures, which expands to TM the well known S ²-twistor bundle of a quaternionic Hermitian manifold M.     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

UNIFORM ESTIMATES OF SOLUTIONS OF δ^--EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for （p, q） differential forms on an open set with C^1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of δ^--equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of δ^--equation for （p, q） differential forms on a Stein manifold.     

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

k-Lucas numbers and associated bipartite graphs

For a positive integer k2, the k-Fibonacci sequence {g_n^(k)} is defined as: g₁^(k)==g_k−2^(k)=0, g_k−1^(k)=g_k^(k)=1 and for n>k2, g_n^(k)=g_n−1^(k)+g_n−2^(k)++g_n−k^(k). Moreover, the k-Lucas sequence {l_n^(k)} is defined as l_n^(k)=g_n−1^(k)+g_n+k−1^(k) for n1. In this paper, we consider the relationship between g_n^(k) and l_n^(k) and 1-factors of a bipartite graph.     

On the relative connections

Abstract

We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Kähler.     

A note on the deformed Hermitian Yang-Mills PDE

We prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob–Yau to obtain an existence result and a priori estimates for some ranges of the phase angle assuming the existence of a subsolution. We then generalise a theorem of Collins–Szèkelyhidi on toric varieties and use it to address a conjecture of Collins–Jacob–Yau.     

Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (M ^d, g) with volume densityθ _m(r)=sinh^d-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M ^2d, g) with volume densityθ _m(r)=sinh^2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.     

Compact complex threefolds which are Kähler outside a smooth rational curve

Let M be a compact complex manifold containing an irreducible curve C such that M — C is Kähler; in this paper we study the link between some cohomological properties of C and the obstructions to the existence of a Kähler metric on the whole of M. In particular we get that, if M is not Kähler, then C is a $ \left({\partial + \bar \partial} \right) $ –exact current, or there exists a positive current S of bidimension (1, 1) such that $ \partial \bar \partial S = 0,\,\chi _C S = 0 $ and S + C is $ \left({\partial + \bar \partial} \right) $ –exact. If C is a smooth rational curve, more precise results are given in connection with the normal bundle N_C|M.     

Graph spectra

The k-spectrum s_k(G) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with s_k (G) = {0, 1, …,(₂^k)} and consider the more general question of describing those sets S {0,1, … ,(₂^k)} such that S = s_k(G) for some graph G.     

Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies

Every closed nanorientable 3-manifold M can be obtained as the union of three orientable handlebodies V₁, V₂, V₃ whose interiors are pairwise disjoint. If g_i denotes the genus of V_i, g₁g₂g₃, we say that M has tri-genus (g₁, g₂, g₃), if in terms of lexicographical ordering, the triple (g₁, g₂, g₃) is minimal among all such decompositions of M into orientable handlebodies. We relate the tri-genus of M to the genus of a surface that represents the dual of the first Stiefel-Whitney class of M. This is used to determine g₁ and g₂.     

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 ＝ {∞} and S2 ＝ {w: Ps(w)＝ 0}, Ps(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f-1(Si) ＝ g-1(Si)(i ＝ 1,2), where f-1(Si) and g-1(Si) denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.     

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