Volume 8, Issue 1

We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g −polystable if and only if it is g −Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X , then a twisted holomorphic vector bundle on X is g −semistable if and only if it is approximate g −Hermite-Einstein.

Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

In this paper we obtain a formula for the number of rational degree d curves in ℙ 3 having a cusp, whose image lies in a ℙ 2 and that passes through r lines and s points (where r + 2 s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ 2 , which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ 3 with one cusp (for δ ≤ 2).

In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.

Let M be a smooth manifold and D = ℒ Ψ +𝒥 Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D * ( M ) of differential forms on M , where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒ Ψ , 𝒥 Ψ are the d * and i * components of D . Under the hypothesis that IdT ( M ) + Ψ is invertible we prove that Ψ=b(Ψ)=-12_(IdTM+Ψ)-1∘[ Ψ,Ψ ]ℱ𝒩 {\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}} , where [·, ·] 𝒡𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e Ψ = ℒ Ψ +𝒥 b ( Ψ ) of the Maurer-Cartan equation according to their type: e Ψ is of finite type r if there exists r ∈ 𝒩 such that Ψr ∘ [ Ψ , Ψ ] 𝒡𝒩 = 0 and r is minimal with this property, where [·, ·] 𝒡𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂ TM of codimension k ⩾ 1 is integrable if and only if the canonical solution e Ψ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.

We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

Let S be a compact complex surface in class VII 0 + containing a cycle of rational curves C = ∑ D j . Let D = C + A be the maximal connected divisor containing C . If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component A i of A is a chain of rational curves which intersects a curve D j of the cycle and for each curve D j of the cycle there at most one chain which meets D j . In other words, we do not prove the existence of curves other those of the cycle C , but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

We introduce an invariant on the Fano polytope of a toric Fano manifold as a polar dual counterpart to the momentum of its polar dual polytope. Moreover, we prove that if the momentum of the polar dual polytope is equal to zero, then the dual invariant on a Fano polytope vanishes.

We study some estimates for a real-valued smooth function φ on almost Hermitian manifolds. In the present paper, we show that ∂∂∂̄ φ and ∂̄∂∂̄ φ can be estimated by the gradient of the function φ .

We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics { ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds { M t } t .

We give several explicit examples of compact manifolds with a 1-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the C 0 -norm. The 4-dimensional examples possess no complex structure, whereas the 6-dimensional example does not possess a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

We study the transverse Kähler holonomy groups on Sasaki manifolds ( M , S ) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b 1 ( M ) and the basic Hodge number h 0,2 B ( S ) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S -unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S -stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U ( n 1 ) × U ( n 2 )) as well as the fiber join operation preserve S -stability.

Let ( M , D ) be a compact contact manifold with dim R M = 2 n ≥ 5. This means that: M is a C ∞ differential manifold with dim R M = 2 n ≥ 5. And D is a subbundle of the tangent bundle TM which satisfying; there is a real one form θ such that D = { X : X ∈ TM, θ ( X ) = 0}, and θ ^ Λ n −1 ( d ) ≠ 0 at every point of p of M . Especially, we assume that our D admits almost CR structure,( M , S ). In this paper, inspired by the work of Matsumoto([M]), we study the difference of partially integrable almost CR structures from actual CR structures. And we discuss partially integrable almost CR structures from the point of view of the deformation theory of CR structures ([A1],[AGL]).

Let 𝒯 s,p,n be the canonical blow-up of the Grassmann manifold G ( p , n ) constructed by blowing up the Plücker coordinate subspaces associated with the parameter s . We prove that the higher cohomology groups of the tangent bundle of 𝒯 s,p,n vanish. As an application, 𝒯 s,p,n is locally rigid in the sense of Kodaira-Spencer.

This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.

In this short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on the total space of the bundle gerbe, just like Kaluza-Klein theory is set on the total space of a principal bundle. This correspondence provides a higher geometric interpretation for para-Hermitian geometry which opens the door to its generalisation to Exceptional Field Theory. This review is based on, but not limited to, my talk at the workshop Generalized Geometry and Applications at Universität Hamburg on 3rd of March 2020.

We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L ∞ -algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.