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Language models (LMs) encode world knowledge in their internal parameters through training. However, LMs may learn personal and confidential information from the training data, leading to privacy concerns such as data leakage. Therefore, research on knowledge deletion from LMs is essential. This study focuses on the knowledge stored in LMs and analyzes the relationship between the side effects of knowledge deletion and the entities related to the knowledge. Our findings reveal that deleting knowledge related to popular entities can have catastrophic side effects. Furthermore, this research is the first to analyze knowledge deletion in models trained on synthetic knowledge graphs, indicating a new direction for controlled experiments.

We prove an index formula for the Dirac operator acting on two-valued spinors on a $3$-manifold $M$ which branch along a smoothly embedded graph $\Sigma \subset M$, and with respect to a boundary condition along $\Sigma$ inspired by an instance of this setting related to the deformation theory of $\mathbb Z_2$-harmonic spinors. When $\Sigma$ is a smooth embedded curve, this index vanishes; this was proved earlier by one of us, but the proof here is different and extends to the more general setting where $\Sigma$ also has vertices. We focus primarily on the Dirac operator itself, but also show how our results apply to more general twisted Dirac operators and to the closely related $\mathbb Z_2$ harmonic $1$-forms.

We collect a number of elementary constructions of $\mathbb Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $\Sigma$ of a $\mathbb Z_2$ harmonic 1-form may exhibit the following features: i) $\Sigma$ may be a non-trivial link; ii) $\Sigma$ may be a multiple cover; iii) $\Sigma$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\mathbb Z_2$ harmonic $1$-forms whose branching sets $\Sigma$ have tangent cones filling out a positive dimensional space, even modulo isometries., Comment: v3: exposition improved

We showed that a sequence of ALH*-gravitational instantons from pairs consisting of a weak del Pezzo surface and a smooth anti-canonical divisor towards a large complex structure limit introduced by Collins, Jacobs and the first author collapsing to a punctured plane with a special Kahler metric, which can be viewed as a non-compact version of the collapsing result of Gross-Wilson. We provide a partial compactification of the moduli space of pointed ALH*-gravitational instantons with respect to the pointed Gromov-Hausdorff topology and locally is a polyhedron complex., Comment: 23 Pages, Comments are welcome. Section 3.3 is added to clarify the cases when singular fibres converge at the large complex structure limit. Some corrections in Section 5 are made

In this paper, we study the modified $J$-equation introduced by Li-Shi. We show that the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional on general compact K\"ahler manifolds. On toric manifolds we give a numerical necessary and sufficient condition for the existence of solutions, extending the results of Collins-Sz\'ekelyhidi., Comment: 38 pages

We introduce the $J$-equation on holomorphic vector bundles over compact K\"ahler manifolds and investigate some fundamental properties as well as examples of them. In particular, we give algebraic obstructions called the (asymptotic) $J$-stability in terms of subbundles on compact K\"ahler surfaces, and a numerical criterion on vortex bundles via dimensional reduction. Also, we discuss an application for the vector bundle version of the deformed Hermitian-Yang-Mills equation in the small volume regime., Comment: 52 pages, made some corrections, Theorem 1.4 (3) is extended for sufficiently smooth vector bundles

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact K\"ahler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song, which makes further progress toward Collins-Jacob-Yau's original conjecture in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case., Comment: 43 pages, added new references and Corollary 1.5 on Collins-Jacob-Yau's conjecture for projective manifolds

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated $(1,1)$-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a $C$-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch., Comment: 24 pages, final version, to appear in Internat. J. Math

We study the line bundle mean curvature flow on K\"ahler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of K\"ahler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian-Yang-Mills equation away from a finite number of curves of negative self-intersection on the surface. As an application, we obtain a lower bound of a Kempf-Ness type functional on the space of potential functions satisfying the hypercritical phase condition., Comment: 19 pages, final version, to appear in Calc. Var. Partial Differential Equations

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the K\"ahler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on K\"ahler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the $H$-functional., Comment: 11 pages, final version, to appear in Proc. Amer. Math. Soc

We study the quantization of coupled K\"ahler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence., Comment: 37 pages, final version, to appear in Analysis & PDE

In this paper, we introduce the "coupled Ricci iteration", a dynamical system related to the Ricci operator and twisted K\"ahler-Einstein metrics as an approach to the study of coupled K\"ahler-Einstein (CKE) metrics. For negative first Chern class, we prove the smooth convergence of the iteration. For positive first Chern class, we also provide a notion of coercivity of the Ding functional, and show its equivalence to existence of CKE metrics. As an application, we prove the smooth convergence of the iteration on CKE Fano manifolds assuming that the automorphism group is discrete., Comment: 18 pages, final version