Study of geometric invariants in Mukai threefolds with C∗ action.
problem Investigate geometric invariants of Mukai threefolds with C∗ action. method Find invariant divisors in the anticanonical system, describe threefolds in terms of quartic, and establish Kähler-Einstein manifolds.
result Established a bound on log canonical thresholds and found additional symmetry in threefolds.
The paper proves K-stability of special Gushel-Mukai manifolds.
problem Proving K-stability of special Gushel-Mukai manifolds.
method Analyzing the structure of Gushel-Mukai manifolds and their K-stability.
result General special Gushel-Mukai n-folds are K-stable for 3 ≤ n ≤ 6.
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.
Study shows Futaki invariant vanishes on most Fano threefolds.
problem Analyzing the Futaki invariant on K-polystable Fano threefolds.
method Examining the Futaki invariant as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group.
result Futaki invariant vanishes identically on most Fano threefolds.
We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian Gr(2,5)⊂P9 by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in P(1,1,1,2,3). We also show that a global log canonical threshold of the Mukai--Umemura variet…
Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
problem Understanding the boundary of K-moduli of prime Fano threefolds of genus twelve.
method Developed a modular relation between Fano threefolds and their anticanonical K3 surfaces, proving forgetful morphism is an open immersion.
result Proved the boundary of K-moduli of V22 is purely divisorial and consists of four irreducible components. 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.
Differential geometry applied to Mukai duality on K3 surfaces.
problem Understanding Mukai duality on K3 surfaces.
method Differential geometric approach.
result New insights into Mukai duality on K3 surfaces.
The paper applies Mukai duality to K3 surfaces to relate G2 structures.
problem Exploring G2 structures on K3 fibrations. method Using adiabatic limits and Mukai duality on K3 surfaces.
result Establishes a mathematical connection between different G2 fibrations. In this short note, we show that given a special Kähler-Einstein degeneration with bounded geometry, for any noncentral fiber, there exists a Kähler-Ricci flow which converges to the Kähler-Einstein metric of the central fiber. As an example, Tian's deformations \cite{Tian97} of the Mukai 3-fold admit Kähler-Ricci flow…
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
Abstract: Analogs of Hilbert-Chow morphism for generalized discriminants.
problem No specific problem stated in the abstract.
method No specific method stated in the abstract.
result No specific key result stated in the abstract.
In this paper we will describe an approach to mirror symmetry for appropriate 1-dimensional DM stacks of arithmetic genus g≤1, called tcnc curves, which was developed by the author with Treumann and Zaslow in arXiv:1103.2462 . This involves introducing a conjectural sheaf-theoretic model for the Fukaya category …
Given two compact hyperkähler surfaces X and Y and a holomorphic vector bundle Q on X×Y, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on X to vector bundles on Y. If X and Y are dual complex tori, this transform …
Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners…
Alternative definition of dDT connections for Spin(7) manifolds.
problem Defining deformed Donaldson-Thomas connections for Spin(7) manifolds.
method Real Fourier-Mukai transform applied to compute new connections.
result Alternative definition of dDT connections is compatible with existing connections.
Paper proves bijection of periodic instantons to singular monopoles.
problem Establishing bijection between spatially periodic instantons and singular monopoles.
method Uses Nahm transform and Fourier-Mukai transform, intertwining with Kobayashi-Hitchin correspondences.
result Nahm transform is a bijection as suggested by heuristic.
The paper classifies diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
problem Classifying diffeomorphism classes of Calabi-Yau threefolds constructed by gluing methods.
method Using differential-geometric gluing method and classifications of simply-connected 6-manifolds.
result Any two doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic…
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
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.
Study proves most Fano threefolds are Kähler-Einstein.
problem Existence of Kähler-Einstein metrics on smooth Fano threefolds.
method Investigated smooth Fano threefolds of Picard rank one and anticanonical degree 22 with C∗ action. result Proved existence of Kähler-Einstein metrics on most such threefolds, except possibly two cases.
We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.
Study noncommutative deformations of Calabi-Yau threefolds.
problem Understanding the geometry of Calabi-Yau threefolds under noncommutative deformations.
method Analyzing the influence of Poisson structures on quantum moduli spaces.
result The choice of Poisson structure significantly affects the geometry of quantum moduli spaces.
The study examines non-Kähler threefolds with specific metrics and finds they are quasi-bundles over surfaces.
problem Understanding non-Kähler threefolds with algebraic dimension two.
method Examining compact complex non-Kähler threefolds with locally conformally Kähler metrics and proving they are quasi-bundles over projective surfaces under certain assumptions.
result They are blown-up quasi-bundles over a projective surface.
Study properties of algebraic threefolds with specific singularities.
problem Characterize and classify algebraic threefolds with certain singularities.
method Use topological invariants, rational homology, Poincaré duality, and Lie algebras.
result Relate topological invariants to Lie algebras and representations.
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.
The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.
Constructs Lagrangians in Calabi-Yau threefolds using tropical curves.
problem Constructing Lagrangian structures in Calabi-Yau threefolds.
method Tropical curves and toric degeneration techniques.
result Constructs multiple Lagrangian rational homology spheres with specific weights.
Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.
problem Existence and properties of very stable Higgs bundles.
method Bialynicki-Birula theory, C∗-actions, Hecke transformations, Fourier-Mukai transforms. result Precise formula for multiplicity of very stable components of global nilpotent cone.
Paper generalizes connections between Lagrangian submanifolds and Yang-Mills connections on tropical manifolds.
problem Generalizing connections between Lagrangian submanifolds and Yang-Mills connections on tropical manifolds.
method Proposed data to glue constructions on tropical manifolds, proving the correspondence without simplifying assumptions.
result Generalized correspondence between Lagrangian submanifolds and Yang-Mills connections on tropical manifolds.
We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192,…
Holomorphic forms found on 3D spaces without zeros.
problem Existence of holomorphic one-forms without zeros on threefolds.
method Classification of real 6-manifolds fibering over the circle.
result Proves a conjecture of Kotschick in dimension three.
We give an elementary argument to compute the α-invariant of this Fano 3-fold, which implies the existence of a Kahler-Einstein metric.
The paper bounds Chern numbers of threefolds, generalizing previous work.
problem Bounding Chern numbers of threefolds, especially those with negative Kodaira dimension.
method Analyzing smooth Mori fibre spaces and applying topological bounds.
result Chern numbers of threefolds are bounded by underlying topological manifolds.
The paper shows how to approximate Kähler threefolds with zero Kodaira dimension.
problem Approximating Kähler threefolds with zero Kodaira dimension.
method Analyzes canonical singularities and uses semiuniversal deformation spaces.
result Every Kähler threefold of Kodaira dimension zero admits small projective deformations.
Proves K-stability of cubic threefolds and calculates their Kähler-Einstein metrics.
problem K-stability of cubic threefolds and explicit calculation of Kähler-Einstein metrics.
method Detailed study of three-dimensional canonical and terminal singularities, estimate of Kawamata log terminal volumes.
result All smooth cubic threefolds admit Kähler-Einstein metrics, and a precise list of singular KE ones is provided.
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …
Visible Lagrangians in Hitchin systems are studied for pillowcase covers.
problem Visible Lagrangians in Hitchin systems intersecting non-trivially.
method Computation of Fourier-Mukai transforms and study of mirror dual branes.
result Mirror dual branes are closely related to Hausel's toy model.
Study on balanced Hermitian threefolds with parallel Bismut torsion.
problem Characterizing compact, balanced BTP threefolds.
method Detailed description of all compact, balanced BTP threefolds.
result Characterization of all compact, balanced BTP threefolds.
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
The paper proves a blow-up formula for Bott-Chern cohomology and shows the bimeromorphic invariance of non-Kählerness degrees on threefolds.
problem The bimeromorphic invariance of the ∂∂ˉ-Lemma on threefolds. method Proves a blow-up formula for Bott-Chern cohomology and applies it to show bimeromorphic invariance of non-Kählerness degrees.
result The non-Kählerness degrees are bimeromorphic invariants on compact complex threefolds, implying the bimeromorphic invariance of the ∂∂ˉ-Lemma. The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.
problem Understanding non-Kähler SYZ mirrors for solvmanifolds and their cohomological properties.
method Investigates geometric and cohomological properties, proving relationships and constructing mirror pairs.
result Proves the Fourier-Mukai transform exchanges type-A and type-B cycles, and provides criteria for non-Kähler SYZ mirror pairs.
Kähler-Ricci flow on threefolds collapses in finite time.
problem Finite time collapse of Kähler-Ricci flow on threefolds.
method Analysis of solutions to Kähler-Ricci flow on compact threefolds.
result If the flow encounters a finite time collapsing singularity, the manifold admits a Fano fibration.
Found a stable 3D shape with specific properties.
problem Finding K-stable Fano threefolds.
method Analyzing specific Fano threefolds with given properties.
result Identified a K-stable Fano threefold with Picard rank 3 and anti-canonical degree 28.