Related papers: The Calabi conjecture and K-stability

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular…

In this note we identify the leading terms of the (reduced) K-energy map with a universal linear combination of the principal and subdominant coefficients of the weight of the $mth$ Hilbert point. This shows that the weight…

We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-H\"{u}bsch duality. Our method is a variant of…

We discuss two closely related Calabi-Yau theorems for degenerations of compact K\"ahler manifolds. The first is a Calabi-Yau theorem for big test configurations, that generalizes a result in [WN24]. It follows from recent joint work with…

We explicitly describe, in the language of four-dimensional N=1 supersymmetric field theory, what happens when the moduli of a heterotic Calabi-Yau compactification change so as to make the internal non-Abelian gauge fields…

In this paper we develop an analogue of the Berkovich analytification for non-necessarily algebraic complex spaces. We apply this theory to generalize to arbitrary compact K\"ahler manifolds a result of Chi Li, proving that a stronger…

The notion that the geometry of spacetime is given by the moduli space of 0-branes is examined in four examples of Calabi-Yau threefolds. An important consideration when determining the moduli space of D-branes is the stability condition…

The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case,…

We identify the stable surfaces around the stable limit of the examples of Y. Lee and J. Park [LP07], and H. Park, J. Park and D. Shin [PPS09] using the explicit 3-fold Mori theory in [HTU13]. These surfaces belong to the…

Moduli stabilisation in superstring compactifications on Calabi-Yau orientifolds remains a key challenge in the search for realistic string vacua. In particular, odd moduli arising from the reduction of 2-forms $(B_2,C_2)$ in type IIB are…

We give an overview of recent work on Dirichlet branes on Calabi-Yau threefolds which makes contact with Kontsevich's homological mirror symmetry proposal, proposes a new definition of stability which is appropriate in string theory, and…

A normal variety $X$ is called Calabi-Yau if $K_X \sim_{\mathbb Q} 0$. The index of $X$ is the smallest positive integer $m$ so that $m K_X \sim 0$. We construct smooth, projective Calabi-Yau varieties in every dimension with doubly…

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau…

The tadpole conjecture suggests that the complete stabilization of complex structure deformations in Type IIB and F-theory flux compactifications is severely obstructed by the tadpole bound on the fluxes. More precisely, it states that the…

Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a K\"ahler form $\omega$ and a holomorphic four-form $\Omega$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the…

We analyze the classical stability of string cosmologies driven by the dynamics of orientifold planes. These models are related to time-dependent orbifolds, and resolve the orbifold singularities which are otherwise problematic by…

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This…

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex manifold of hyperkaehler type is Mirror…

We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical…

This article generalises to K\"ahler orbifolds general results on uniformisation of compact K\"ahler manifolds such as the Shafarevich conjecture for linear fundamental groups.