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  1. ; ; (, Journal of Algebraic Geometry)
    The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities x ∈ ( X , Δ ) x\in (X,\Delta ) satisfies the ACC if the coefficients of Δ \Delta belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of δ \delta -plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities. 
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  2. ; (, Bulletin of the London Mathematical Society)
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  3. ; ; (, Journal of the Institute of Mathematics of Jussieu)
    Abstract We show that the K-moduli spaces of log Fano pairs $$\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $$\mathbb {P}^3$$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $$\mathbb {P}^1\times \mathbb {P}^1$$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces. 
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  4. ; ; ; (, Selecta Mathematica)
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