In this article we prove the existence of Kähler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings. P N by sections of K −λ X . A weak Kähler-Einstein metric on X is a Kähler current in 2πc 1 (X) with locally continuous potential, and that is a smooth Kähler-Einstein metric on the smooth part X reg of X. Note there are different definitions in the literature, but they are all equivalent in this context [23]. A Q-Fano variety X is called Q-Gorenstein smoothable 1 if there is a flat family π : X → ∆, over a disc ∆ in C so that X ∼ = X 0 , X t is smooth for t = 0, and X admits a relatively Q-Cartier anti-canonical divisor −K X /∆ (in this case π : X → ∆ is called a Q-Gorenstein smoothing of X 0 ). It is well-known that, by possibly shrinking ∆, we may assume that X t is a Fano manifold for t = 0, and that there exists an integer λ > 0 such that K −λ Xt are very ample line bundles with vanishing higher cohomology for all t ∈ ∆. Moreover, the dimension, denoted by N (λ), of the corresponding linear systems | − λK Xt | is constant in t. Thus, when needed, we may assume that the family X is relatively very ample, i.e. there is a smooth embedding i :Xt . The main theorem of this paper is the following result, which extends the results of [12,13,14,15] to Q-Gorenstein smoothable Q-Fano varieties and simultaneusly gives some understanding on the way such singular metric spaces are approached by smooth KE metrics on the (analytically) nearby Fano manifolds.Theorem 1.1. Let π : X → ∆ be a Q-Gorenstein smoothing of a Q-Fano variety X 0 . If X 0 is K-polystable then X 0 admits a weak Kähler-Einstein metric ω 0 .Moreover, assuming that the automorphism group Aut(X 0 ) is discrete, X t admit smooth Kähler-Einstein metrics ω t for all |t| sufficiently small and (X 0 , ω 0 ) is the limit in the Gromov-Hausdorff topology of (X t , ω t ), in the sense of [21].Few remarks are in place. First, by the generalized Bando-Mabuchi uniqueness theorem [6] the above weak Kähler-Einstein metric ω 0 is unique up to Aut 0 (X 0 ), the identity component of Aut(X 0 ), thus can be viewed as a canonical metric on X 0 . Second, the above theorem does not just state that there is a sequence of nearby KE Fano manifolds which converges, in the Gromov-Hausdorff sense, to the weak KE metric, but that all the nearby KE Fano manifolds actually converge to the unique singular limit (X 0 , ω 0 ). Thus this property provides a good topological correspondence between complex analytic deformations (alias flat-families) and the notion of Gromov-Hausdorff convergence. For example, by the Bando-Mabuchi theorem we have the following immediate corollary: Corollary 1.2. Let π : X → ∆ and π ′ : X ′ → ∆ be two Q-Gorenstein smoothings of Q-Fano varieties X 0 and X ′ 0 . Suppose X t and X ′ t are bi-holomorphic for all t = 0, and X 0 and X ′ 0 are both K-polystable with discrete automorphism group, then X 0 and X ′ 0 are isomorphic varieties.Notice t...

A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive-definite subspace of Λ 2 for the wedge product. This article is motivated by a conjecture of Donaldson: when X is compact ω can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω on a compact manifold X defines a natural G2-structure φ on X × T 3 which has vanishing torsion precisely when ω is a hyperkähler triple. We study the G2-Laplacian flow starting from φ, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding G2-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow). Preliminary definitions and short time existenceThe main result of this section is:Proposition 2.1. Let ω be a hypersymplectic structure on a compact 4-manifold X. Then there exists a unique short time solution ω(t) to the hypersymplectic flow (3) starting at ω.This will follow easily from the analogous result of Bryant-Xu for the G 2 -Laplacian flow. Along the way we make explicit the relationship between the natural metric g ω on X induced by a Proof. This is immediate from the uniqueness part of Bryant-Xu's theorem [4, Theorem 0.1]: since the initial data φ(0) is T 3 invariant, so is the ensuing flow.By T 3 -invariance, the 3-form φ(t) on X × T 3 necessarily has the shape∈ Ω 2 (X) and D(t) ∈ Ω 3 (X) are paths of forms on X. Moreover, since dφ(t) = 0, it follows that these forms on X are closed.Lemma 2.7. A(t) = 1.Proof. We know that dA(t) = 0, i.e., that for every t, A(t) is constant. Moreover,Since ∂ t φ is exact, this integral is independent of t, and so A(t) = A(0) = 1.Next we consider the involution ϑ : X × T 3 → X × T 3 given by ϑ(p, t) → (p, −t) and writê φ(t) = −ϑ * φ(t).Lemma 2.8. For all t,φ(t) = φ(t). Hence B i (t) = 0 = D(t) vanish identically, and φ(t) remains of the form (2) for as long as it exists, for a closed triple ω(t) of 2-forms on X.

The existence of weak conical Kähler-Einstein metrics along smooth hypersurfaces with angle between 0 and 2π is obtained by studying a smooth continuity method and a local Moser's iteration technique. In the case of negative and zero Ricci curvature, the C 0 estimate is unobstructed; while in the case of positive Ricci curvature, the C 0 estimate obstructed by the properness of the twisted K-Energy. As soon as the C 0 estimate is achieved, the local Moser iteration could improve the rough bound on the approximations to a uniform C 2 bound , thus produce a weak conical Kähler-Einstein metric. The method used here do not depend on the bound of any background conical Kähler metrics.

We prove the hypersymplectic flow of simple type on standard torus T4 exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity‐one G2‐Laplacian flow on a compact 7‐manifold which exists for all time and converges to a torsion‐free G2 structure modulo diffeomorphisms.

The continuity method is used to deform the cone angle of a weak conical K\"ahler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regularity result of Guenancia-P$\breve{\text{a}}$un \cite{GP} and Chen-Wang \cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical K\"ahler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a \emph{weak conical K\"ahler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).Comment: Theorem 1.3 of version 1 removed due to imprecise statement; English grammar mistakes and typos corrected, missing reference adde