On the existence of balanced metrics on six-manifolds of cohomogeneity one

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler $$\text {SU}(3)$$ SU ( 3 ) -structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.

Solutions to the Hull–Strominger System with Torus Symmetry

We construct new smooth solutions to the Hull–Strominger system, showing that the Fu–Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $$13 \le k \le 22$$ 13 ≤ k ≤ 22 and $$14\le r\le 22$$ 14 ≤ r ≤ 22 , the smooth manifolds $$S^1\times \sharp _k(S^2\times S^3)$$ S 1 × ♯ k ( S 2 × S 3 ) and $$\sharp _r (S^2 \times S^4) \sharp _{r+1} (S^3 \times S^3)$$ ♯ r ( S 2 × S 4 ) ♯ r + 1 ( S 3 × S 3 ) , have a complex structure with trivial canonical bundle and admit a solution to the Hull–Strominger system.

Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)

We investigate the deformations of pairs [Formula: see text], where [Formula: see text] is a line bundle on a smooth projective variety [Formula: see text], defined over an algebraically closed field [Formula: see text] of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair [Formula: see text] is homotopy abelian whenever [Formula: see text] has trivial canonical bundle, and so these deformations are unobstructed.

Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

Ordinary varieties with trivial canonical bundle are not uniruled

We prove that smooth, projective, K-trivial, weakly ordinary varieties over a perfect field of characteristic $$p>0$$ p > 0 are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer’s results, implies that varieties of the above type have strongly semistable tangent bundles with respect to every polarization.

Resolution à la Kronheimer of $$\mathbb {C}^3/\Gamma $$ singularities and the Monge–Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds $$Y^\Gamma $$ Y Γ that are supposed to be the crepant resolution of quotient singularities $$\mathbb {C}^3/\Gamma $$ C 3 / Γ with $$\Gamma $$ Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms $$\omega ^{2,1}$$ ω 2 , 1 . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on $$Y^\Gamma $$ Y Γ . We study the issue of Ricci-flat Kähler metrics on such resolutions $$Y^\Gamma $$ Y Γ , with particular attention to the case $$\Gamma =\mathbb {Z}_4$$ Γ = Z 4 . We advance the conjecture that on the exceptional divisor of $$Y^\Gamma $$ Y Γ the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on $$\mathrm{tot} K_{{\mathbb {W}P}[112]}$$ tot K W P [ 112 ] that we construct, i.e., the total space of the canonical bundle of the weighted projective space $${\mathbb {W}P}[112]$$ W P [ 112 ] , which is a partial resolution of $$\mathbb {C}^3/\mathbb {Z}_4$$ C 3 / Z 4 . For the full resolution, we have $$Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}$$ Y Z 4 = tot K F 2 , where $$\mathbb {F}_2$$ F 2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $$\mathbb {F}_2$$ F 2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.

A rigid, not infinitesimally rigid surface with K ample

We produce an example of a rigid, but not infinitesimally rigid smooth compact complex surface with ample canonical bundle using results about arrangements of lines inspired by work of Hirzebruch, Kapovich & Millson, Manetti and Vakil.