Curvature constraints in heterotic Landau-Ginzburg models

In this paper, we study a class of heterotic Landau-Ginzburg models. We show that the action can be written as a sum of BRST-exact and non-exact terms. The non-exact terms involve the pullback of the complexified Kähler form to the worldsheet and terms arising from the superpotential, which is a Grassmann-odd holomorphic function of the superfields. We then demonstrate that the action is invariant on-shell under supersymmetry transformations up to a total derivative. Finally, we extend the analysis to the case in which the superpotential is not holomorphic. In this case, we find that supersymmetry imposes a constraint which relates the nonholomorphic parameters of the superpotential to the Hermitian curvature. Various special cases of this constraint have previously been used to establish properties of Mathai-Quillen form analogues which arise in the corresponding heterotic Landau-Ginzburg models. There, it was claimed that supersymmetry imposes those constraints. Our goal in this paper is to support that claim. The analysis for the nonholomorphic case also reveals a constraint imposed by supersymmetry that we did not anticipate from studies of Mathai-Quillen form analogues.

Lagrangian foliations and Anosov symplectomorphisms on Kähler manifolds

We investigate parallel Lagrangian foliations on Kähler manifolds. On the one hand, we show that a Kähler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kähler manifolds which are not tori. These examples arise from Anosov automorphisms preserving a Kähler form.

Partially regular and cscK metrics

A Kähler metric [Formula: see text] with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to [Formula: see text] is a positive constant for all integer [Formula: see text] sufficiently large. The aim of this paper is to prove that for all [Formula: see text] there exists an [Formula: see text]-dimensional complex manifold equipped with strictly partially regular and cscK metric [Formula: see text]. Further, for [Formula: see text], the (constant) scalar curvature of [Formula: see text] can be chosen to be zero, positive or negative.

Kähler–Ricci flow on homogeneous toric bundles

Assume that [Formula: see text] is a homogeneous toric bundle of the form [Formula: see text] and is Fano, where [Formula: see text] is a compact semisimple Lie group with complexification [Formula: see text], [Formula: see text] a parabolic subgroup of [Formula: see text], [Formula: see text] is a surjective homomorphism from [Formula: see text] to the algebraic torus [Formula: see text], and [Formula: see text] is a compact toric manifold of complex dimension [Formula: see text]. In this note, we show that the normalized Kähler–Ricci flow on [Formula: see text] with a [Formula: see text]-invariant initial Kähler form in [Formula: see text] converges, modulo the algebraic torus action, to a Kähler–Ricci soliton. This extends a previous work of Zhu. As a consequence, we recover a result of Podestà–Spiro.

Quaternion-Kähler manifolds near maximal fixed point sets of $$S^1$$-symmetries

Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.

Geometric quantization of the Hitchin system

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.

Deligne pairing and Quillen metric

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

Complex Hessian Equations on Some Compact Kähler Manifolds

On a compact connected2m-dimensional Kähler manifold with Kähler formω, given a smooth functionf:M→ℝand an integer1<k<m, we want to solve uniquely in[ω]the equationω̃k∧ωm-k=efωm, relying on the notion ofk-positivity forω̃∈[ω](the extreme cases are solved:k=mby (Yau in 1978), andk=1trivially). We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.

A NOTE ON MOMENT MAPS AND KÄHLER–EINSTEIN MANIFOLDS

We collect some properties of the moment map relative to the isometric and holomorphic action of a compact Lie group G on a (compact) Kähler (Einstein) manifold; in particular, we study some invariants which only depend on the cohomology class of the invariant Kähler form and then specialize to the complex projective space when the group G is simple and acts linearly.