Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.
problem Existence of Kähler-Ricci soliton degeneration for log Fano pairs.
method Algebraic proof of two-step degeneration to uniformly Ding stable triple.
result Log Fano pairs admit Kähler-Ricci soliton when ground field is complex.
Proves unique degeneration of log Fano fibration germs.
problem Stable degeneration of log Fano fibration germs.
method Introduced the H-invariant for filtrations over log Fano fibration germs and used a unique quasi-monomial valuation to achieve the degeneration.
result Unique K-polystable special degeneration of log Fano fibration germs.
Study real semi-stable degenerations and describe real loci via blow-ups.
problem Describe the homeomorphism type of real loci in degenerations.
method Use real-oriented blow-ups to describe the homeomorphism type of real loci.
result Give more explicit descriptions of real loci as stratified spaces.
Classifies normal stable Horikawa surfaces with smoothable singularities.
problem Characterizing surfaces with specific singularities and smoothability criteria.
method Classification and smoothability criterion based on log canonical singularities.
result Provides a criterion for global Q-Gorenstein smoothability of Horikawa surfaces. Stable solutions found for a specific physics model.
problem Stability of solutions to the U(1)-Yang-Mills-Higgs model. method Gluing method and detailed analysis of linearized operators.
result Found a family of stable critical points in higher dimensions.
We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to ob…
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
Proves a conjecture for Calabi-Yau manifolds.
problem Maximal degeneration of Calabi-Yau manifolds.
method Valuative independence condition for section ring.
result Metric SYZ conjecture proven.
We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension a…
Study explores unstable 3-forms on Calabi-Yau 3-folds.
problem Understanding degenerations of Calabi-Yau 3-folds via 3-forms.
method Investigates geometries of 3-forms on symplectic 6-manifolds.
result Unstable 3-forms reveal rich geometric properties related to SYZ conjecture.
In this paper, we introduce some notions on the pair consisting of a Chern connection and a Higgs field closely related to the first and second variation of Yang-Mills- Higgs functional, such as strong Yang-Mills-Higgs pair, degenerate Yang-Mills-Higgs pair, stable Yang-Mills-Higgs pair. We investigate some properties …
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Sharp diameter bounds for Calabi-Yau degenerations proved.
problem Bounding the diameter of Calabi-Yau metrics during degeneration.
method Sharp upper and lower bounds derived for Ricci-flat Kahler metrics.
result Conjecture confirmed by obtaining precise diameter bounds.
We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
problem Proving a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
method Analyzing degenerating families of projective normal varieties and studying the limiting behavior of semistable bundles.
result Improves several previously known algebro-geometric results on normalized tautological classes and proves a new version of the singular Donaldson-Uhlenbeck-Yau theorem.
Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
Paper proves contractible fake surfaces up to complexity 6 are deformable.
problem Stable Andrews-Curtis conjecture and contractible fake surfaces.
method Induction scheme proving contractibility up to complexity 6.
result Contractible fake surfaces up to complexity 6 are 3-deformable.
The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limit…
Extends arguments to limit structure in Calabi-Yau degenerations.
problem Understanding Gromov-Hausdorff limits in degenerating Calabi-Yau manifolds.
method Reduces conjecture to partial second-order estimate.
result Extends arguments to new settings.
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
In this paper, the following three are shown. (1) For a C∞ convex integrand γ:Sn→R+, its dual convex integrand δ:Sn→R+ is of class C∞. (2) For a stable convex integrand γ:Sn→R+, its dual convex integrand δ:Sn→R+ is stable. (3) Let $γ: S…
Degenerate solutions found in 2D H-system bubbles with higher degrees.
problem Existence of degenerate solutions in H-system bubbles with degree ≥ 3.
method Algebraic characterization of degenerate bubbles.
result Degenerate solutions can exist for H-system bubbles with degree ≥ 3.
Proves section conjecture for curves and surface bundles over various fields.
problem Proving Grothendieck's section conjecture for curves and surface bundles.
method Formulated and proved the section conjecture for stable graphs, used Galois cohomology classes to obstruct sections.
result Proved section conjecture for curves and surface bundles over p-adic and number fields.
We prove that for two germs of analytic mappings f,g:(Cn,0)→(Cp,0) with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family {ft} of analyt…
Unique solution found for Demailly's equation on stable bundles.
problem Existence of a Griffiths positively curved metric on Hartshorne ample vector bundles.
method Proved an essentially unique solution to a Hermitian-Einstein-type equation for stable bundles.
result The proposed approach by Demailly must be modified to tackle the conjecture.
We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the…
Establishes bounds on Andrews-Curtis moves for trivial group presentations.
problem Understanding presentations of the trivial group and Andrews-Curtis moves.
method Explicit upper bounds on stable Andrews-Curtis moves for thickenable presentations.
result Thickenable presentations of the trivial group satisfy the Andrews-Curtis conjecture.
Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.
problem Degeneration of minimal hypersurfaces under stability or bounded index constraints.
method Analysis of sequences of minimal hypersurfaces, parameterization with controlled maps, and topological finiteness results.
result Minimal hypersurfaces can degenerate to singular ones with controlled geometry, topology, and singular set.
We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…
New geometric proof shows index of umbilic points on analytic surfaces is at most one.
problem Proving the Carathéodory Conjecture for compact simply connected embedded surfaces.
method Geometric analysis of degenerate umbilic points on analytic surfaces.
result Index of an umbilic on an analytic surface cannot be an integer larger than one.
We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.
Study on metric bubbles in complex dimensions 1 and 2.
problem Understanding degenerations of Kähler-Einstein metrics.
method Investigation of metric bubble trees for non-collapsing cases.
result Description of a conjectural higher-dimensional picture.
Study on stable Hamiltonian topology finds non-density of certain structures.
problem Non-density of stable hypersurfaces and Hamiltonian structures.
method Proving non-density results for stable hypersurfaces and Hamiltonian structures in various dimensions.
result Non-density of stable hypersurfaces and Hamiltonian structures in specific isotopy and homotopy classes.
Study optimal degenerations of Fano threefolds, proving K-polystability and Kähler-Ricci solitons.
problem Optimal degenerations of K-unstable Fano threefolds.
method Explicitly determined degenerations, finding weighted K-polystable (X0,ξ0), studying moduli spaces. result One moduli space is isomorphic to the GIT-moduli space of biconic curves, the other is a single point.
The paper verifies stable handleslide triviality of some R-links and shows many are stably equivalent.
problem Stable handleslide triviality of R-links as potential counterexamples to the generalized property R conjecture.
method Implemented an algorithm to construct all R-links explicitly and verified their stable handleslide triviality.
result Many R-links are stably handleslide equivalent.
Paper proves convex domains have one maximum for semi-stable solutions.
problem Analyzing critical points of semi-stable solutions on convex domains.
method Relating critical points to an auxiliary function and using topological degree.
result Positive, semi-stable solutions have exactly one non-degenerate critical point.
This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…
Unique K-polystable degenerations for Fano varieties confirmed.
problem Algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
method Study of optimal degeneration problems via new functionals of real valuations.
result Confirm algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
Let Y be a closed 3-manifold such that all flat SU(2)-connections on Y are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on Y for stable flat SL(2,C) connections satisfying an L2-bound for the real curvature. Combining the compactness theorem and a previous…
The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs.…
The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
problem Characterizing nonpositively curved 4-manifolds with zero Euler characteristic.
method Analyzing the Ricci curvature and foliations in neighborhoods of points.
result Nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.
Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.