The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
Let X be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants (DT4 invariants) by studying moduli spaces of solutions to the Donaldson-Thomas equations on X. We also study sheaves counting problems on local Calabi-Yau 4-folds. We relate DT4…
For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth t…
Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations …
We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds.…
First steps towards a classification of irreducible symplectic 4-folds whose integral 2-cohomology with 4-tuple cup product is isomorphic to that of Hilb^2(K3). We prove that any such 4-fold deforms to an irreducible symplectic 4-fold of Type A or Type B. A 4-fold of Type A is a double cover of a (singular) sextic hype…
We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative…
Given a coassociative 4-fold N with a conical singularity in a varphi-closed 7-manifold M (a manifold endowed with a distinguished closed 3-form varphi), we construct a smooth family, {N'(t): t\in(0,tau)} for some tau>0, of (smooth, nonsingular,) compact coassociative 4-folds in M which converge to N in the sense of cu…
Given a complex 4-fold X with an (Calabi-Yau 3-fold) anti-canonical divisor Y, we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of Y. We also discuss gluing formulas which relate relative invariants and DT4 invariants for Calabi-Yau 4-folds.
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…
In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the …
Invariant obstructs separating coassociative 4-folds.
problem Obstructing the separation of coassociative 4-folds.
method Defining a Z2-valued invariant for coassociative 4-folds with spin structures. result Invariant provides an obstruction to separating coassociative 4-folds.
Infinite Klein bottles with 4-fold meridians found.
problem Finding indecomposable Klein bottles with specific meridian orders.
method Exhibited an infinite family of Klein bottles in 4-sphere.
result Infinite family of indecomposable Klein bottles with order-4 meridians.
Motivated by Strominger-Yau-Zaslow's mirror symmetry proposal and Kontsevich's homological mirror symmetry conjecture, we study mirror phenomena (in A-model) of certain results from Donaldson-Thomas theory for Calabi-Yau 4-folds.
We address the problem of classification of hyper-Kähler fourfolds with b2=23. In particular we prove some special cases of the Conjecture of O'Grady about hyper-Kähler 4-folds numerically equivalent to the Hilbert scheme of two points on a K3 surface.
We study coassociative 4-folds N in R^7 which are asymptotically conical to a cone C with rate lambda<1. If lambda is in the interval [-2,1) and generic, we show that the moduli space of coassociative deformations of N which are also asymptotically conical to C with rate lambda is a smooth manifold, and we calculate it…
The paper extends orientability results for DT invariants on Calabi-Yau 4-folds.
problem Defining Donaldson-Thomas invariants on quasi-projective Calabi-Yau 4-folds.
method Extending orientability results to quasi-projective Calabi-Yau 4-folds and using gauge theory.
result Orientability of moduli stacks and invariants on quasi-projective Calabi-Yau 4-folds.
Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are bi-Lagrangian and that they are obtained by complex submanifolds via a sort of "hyperkaehler rotation trick"; thus they retain part of the rigidity o…
We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degr…
The paper extends the Hopf differential concept to associative submanifolds in G2-manifolds.
problem Understanding the geometry of associative submanifolds in G2-manifolds.
method Analogy with CMC surfaces in R^3 and use of spinor theory.
result Every non-totally-geodesic associative 3-fold in R^7, T^7, and S^7 admits non-vanishing harmonic twisted spinors.
This paper extends to dimension 4 the results in the article "Second Order Families of Special Lagrangian 3-folds" by Robert Bryant. We consider the problem of classifying the special Lagrangian 4-folds in C^4 whose fundamental cubic at each point has a nontrivial stabilizer in SO(4). Points on special Lagrangian 4-fol…
In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…
We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on…
Kähler-Ricci flow shows type II singularity on Fano threefolds.
problem Understanding the behavior of Kähler-Ricci flow on Fano threefolds.
method Analyzing the Kähler-Ricci flow on Fano threefolds from a specific family.
result Kähler-Ricci flow develops type II singularity on Fano threefolds from the specified family.
It is shown that coassociative cones in R^7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R^7. The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S^6. This leads to an equivalence between associative …
New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
Proves properness of K-moduli spaces for Fano varieties.
problem Proving properness of moduli spaces of K-polystable Fano varieties.
method Algebraic approach, studying test configurations, constructing stratification.
result Proves properness under specific divisorial valuation condition.
New toric Fano manifolds found without extremal Kähler metrics.
problem Finding toric Fano manifolds without extremal Kähler metrics.
method Constructing specific toric Fano manifolds of dimensions 10 and n (n≥11) that do not admit extremal Kähler metrics.
result Existence of toric Fano manifolds of dimension 10 and higher that do not admit extremal Kähler metrics.
Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.
problem Analyzing the limits of Kähler-Ricci flow on Fano G-manifolds.
method Proves the Gromov-Hausdorff limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
result The limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
The study classifies complex smooth Fano varieties with large pseudoindex.
problem Classifying Fano varieties with specific properties.
method Analyzing Fano varieties with large pseudoindex and Picard number greater than one.
result Classification of Fano varieties with pseudoindex at least n-2 and Picard number greater than one.
New approach proves K-stability of Fano varieties.
problem Proving K-stability of Fano varieties.
method Developed a general approach using admissible flags.
result Proved K-stability of smooth Fano hypersurfaces of index two.
K-stability proven for a specific type of Fano threefold.
problem Proving K-stability of Fano threefolds.
method Analyzing double covers of blow-ups with specific branch divisors.
result Proven K-stability of Fano threefolds of rank 2 and degree 14.
Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.
problem Semipositivity and nefness of Chow-Mumford line bundle for K-semistable log-Fano pairs.
method Alternative proof using families of K-semistable log-Fano pairs.
result Proof of semipositivity and nefness for K-semistable log-Fano pairs.
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 calculates volumes of Fano K-moduli spaces in various dimensions.
problem Computing volumes of Fano K-moduli spaces in different dimensions.
method Computed volumes of Fano K-moduli spaces of Quartic del Pezzo and log Fano hyperplane arrangements in dimensions one and two.
result Relates computed volumes to Weil-Petersson volumes, extending the Weil-Petersson metric to the log case.
This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between…
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.
Classifies Fano varieties with large pseudoindex and non-free rational curves.
problem Classifying Fano varieties with specific properties.
method Extremal contractions and classification of varieties.
result Complete classification of Fano n-folds with pseudoindex at least n−2 and Picard number greater than one. Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
We prove the Yau-Tian-Donaldson's conjecture for any Q-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. Proof of flow convergence on Fano manifolds.
problem Convergence of Kahler-Ricci flow on Fano manifolds.
method Recent techniques in geometry and analysis.
result Convergence of Kahler-Ricci flow on Fano manifolds.
The paper proves Mabuchi solitons and constants on Fano admissible manifolds.
problem Existence of Mabuchi solitons on Fano admissible manifolds.
method Defined Mabuchi solitons and constants, proved existence and non-existence.
result Fano admissible manifolds admit Mabuchi solitons if and only if the Mabuchi constant is less than 1.
New examples of Kähler-Ricci solitons on Fano threefolds with non-trivial moduli found.
problem Finding Fano threefolds with Kähler-Ricci solitons and non-trivial moduli.
method Established weighted K-stability and GIT-stability criteria, generalized Koiso's theorem, and developed the weighted Abban-Zhuang estimate.
result First examples of strictly weighted K-semistable Fano varieties and new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups.
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…
New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.
problem Identifying Fano threefolds of degree 22 with Kähler-Einstein metrics.
method Analyzing the structure and properties of Fano threefolds of Picard rank one with anti-canonical degree 22.
result All remaining Fano threefolds of degree 22 admit Kähler-Einstein metrics.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large an…