Search arXivSearch

arXiv · math/9910027

Rational homotopy types of mirror manifolds

Abstract

We explain how to relate the problem of finding a mirror manifold for a Calabi-Yau manifold to the problem of characterizing the rational homotopy types of closed Kähler manifolds.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jian Zhou. 1999-10-05. Rational homotopy types of mirror manifolds. https://arxiv.org/abs/math/9910027

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Mass and rigidity in almost Kähler geometry

We derive an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost Kähler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the underlying almost complex structure, extending a result of Hein and LeBrun in the Kähler ALE case. Our approach is based on a spin$^\mathbb {C}$ adaptation of Witten's proof of the positive mass conjecture in the spin case and is therefore distinct from previous complex-geometric methods. In dimension $4$, we prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost Kähler manifolds. We also study rigidity phenomena of almost Kähler ALE manifolds. We prove that an almost Kähler-Einstein ALE manifold with nonnegative scalar curvature and certain decay assumptions is necessarily Kähler-Einstein. In particular, any four dimensional Ricci-flat almost Kähler manifold with maximal volume growth and curvature in $L^2$ is Kähler, yielding new evidence towards the Bando--Kasue--Nakajima conjecture. We also discuss analogous rigidity results for asymptotically locally flat (ALF) manifolds.

math.DG

A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere

Let $\{Σ_a\}$, $a\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\subset\mathbb{S}^3$, $R(a)>π/2$. We prove that $R$ is real-analytic, tends to $π/2$ at both ends, and therefore folds: it has an interior maximum $R_*>π/2$ and is not injective. Hence each $B(ρ)$ with $π/2<ρ π/2$. The exact identity $\dim K_0^{ev}(Σ_a)=\mathbf{1}_{\{R'=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\partial_ηφ_a-\operatorname{ct}_κ(r(a))φ_a=r'(a)A_a(η,η)$, a Robin defect identity, which requires of the ambient only that the barrier family be umbilic and which extends to capillary boundary conditions at constant contact angle.

math.DG

Demailly-Kollár continuity on klt pairs, and applications to alpha and delta invariants

We establish a Demailly-Kollár type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal Kähler klt pairs, thereby completing a program initiated by the second author. We further show that the two local alpha invariants introduced by Guedj and Trusiani for an isolated log terminal singularity coincide and that their common value is the $n$th root of Li's normalized volume. These identities have geometric consequences. The equality for the delta invariant yields a Yau-Tian-Donaldson type divisorial criterion for the solvability of twisted Kähler-Einstein equations in big cohomology classes, without a semipositivity assumption on the twist. The local alpha identity determines algebraically the critical exponent governing the existence of positively curved KE metrics near an isolated log terminal singularity, thus confirming a prediction of Guedj and Trusiani.

math.DG