Characterizes Wahl singularities in del Pezzo surface degenerations.
problem Classifying Wahl singularities in degenerations of del Pezzo surfaces.
method Introducing del Pezzo Wahl chains with markings, proving degenerations to toric surfaces, establishing correspondences, and using Hacking's exceptional collections.
result Established a one-to-one correspondence between marked del Pezzo surfaces and fake weighted projective planes.
In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with Du Val singularities and of del Pezzo surfaces of Picard rank 1 with Du Val singu…
Classifies K-stable Gorenstein del Pezzo surfaces with solitons.
problem Classifying K-stable Gorenstein del Pezzo surfaces for solitons.
method Analyzes vector fields and stability conditions.
result Provides a classification of K-stable pairs (X,v).
We study global log canonical thresholds of del Pezzo surfaces.
A smooth curve found in a space of special surfaces.
problem Understanding the structure of K-polystable del Pezzo surfaces.
method Explicit construction of a component in the K-moduli space.
result Found a smooth rational curve in the K-moduli space.
We prove functional identities for conic webs on del Pezzo surfaces.
problem Functional identities for conic webs on del Pezzo surfaces.
method Uniform approach based on Weyl group actions and classical results.
result Existence of hyperlogarithmic functional identities of weight 7-d.
We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger than one.
In this note, we compute the Poisson cohomology groups for any Poisson Del Pezzo surface.
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler-Einstein metric. In particular, all del Pezzo surfaces of degree 4,5, or 6 and certain special del Pezzo surfaces of lower degree are…
The paper classifies Fano distributions on specific Fano manifolds.
problem Investigating Fano distributions on Fano manifolds.
method Classification of Fano distributions on various Fano manifolds.
result Classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one.
We prove new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type A1, A2, A3, A4, A5 or $\mathbb{A}_{6…
Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.
problem Understanding real lines on real del Pezzo surfaces of degree 1.
method Intrinsically defined Pin-structure on the real locus.
result The difference between hyperbolic and elliptic lines is always 16.
We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.
We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in P(a1,a2,a3,a4). As a corollary we obtain the existence of orbifold Kähler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-…
We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.
The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.
problem Understanding mirror symmetry for del Pezzo surfaces and related geometric structures.
method Using hyperKähler rotation and Floer theory, the paper constructs and compares Landau-Ginzburg mirrors and complex affine structures.
result The limit of the complex affine structure of special Lagrangian fibrations agrees with integral affine structures.
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose alpha-invariant of Tian is greater than 2/3.
We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the deg…
We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kahler metric in the Kahler class corresponding to the polarization.
Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.
problem Characterizing log del Pezzo surfaces with specific geometric properties.
method Examining two classes of non-toric log del Pezzo surfaces and analyzing their geometric properties.
result Examples found that admit Kähler-Ricci solitons but not Sasaki-Einstein cone links.
The study classifies involutions on del Pezzo surfaces.
problem Classifying involutions on del Pezzo surfaces.
method Mapping class group theory and hyperbolic reflection groups.
result A complete classification of involutions on del Pezzo surfaces.
Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of the complex projective plane in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.
From a hermitian metric on the anticanonical bundle on a Del Pezzo surface, and a holomorphic section of it, we construct a one parameter family of bihermitian metrics (or equivalently generalized Kaehler structures). The construction appears to be linked to noncommutative geometry.
Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.
problem Characterize the real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.
method Explicit description of isotopy types of real lines and presentation of MW group in mapping class group.
result Explicit formula for the action of MW group in H1(XR).
The global log canonical threshold (or Tian's alpha-invariant) plays an important role in the geometry of Fano varieties. Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to determine when a smooth Fano variety X admits…
Researchers compute monodromy groups of surface families over quartic curves.
problem Computing monodromy groups of surface families over smooth quartic curves.
method Analyzing cyclic branched covers of P2 over smooth quartic curves, computing monodromy groups for del Pezzo and K3 surfaces. result Obtained monodromy groups for del Pezzo and K3 surfaces, including Weyl group $W\left(E_{7}
ight)$ and arithmetic lattice $U\left(h_{L_{-}}
ight)$.
Researchers prove mirror symmetry for certain non-compact Calabi-Yau surfaces.
problem Proving mirror symmetry for non-compact Calabi-Yau surfaces.
method Surveying recent works on Strominger-Yau-Zaslow mirror symmetry for rational elliptic surfaces and del Pezzo surfaces.
result All ALH∗ gravitational instantons can be compactified to weak del Pezzo surfaces. Study real algebraic curves on real del Pezzo surfaces using degeneration methods.
problem Classify topological types of real algebraic curves on real del Pezzo surfaces.
method Degeneration methods and real enumerative geometry.
result Obstructions and constructions of real algebraic curves with prescribed topology.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
Study monodromy factorizations for lines on del Pezzo surfaces.
problem Understanding monodromy factorizations for lines on del Pezzo surfaces.
method Listed monodromy factorizations in the mapping class group of a torus.
result Explicit correspondence between factorizations and roots of E8.
The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).
Let (X,J) be a compact Kahler manifold with a non-zero holomorphic Poisson structure β. If the obstruction space for deformations of generalized complex structures on (X,J) vanishes, we obtain a family of deformations of non-trivial bihermitian structures (J,Jt−,ht) on X by using β. In addition, if t…
On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in C∞-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.
Develops techniques to estimate thresholds for logarithmic surfaces, proving existence of Kahler-Einstein metrics.
problem Estimating basis log canonical thresholds on logarithmic surfaces.
method Develops new local intersection estimates to imply log canonicity.
result Shows the existence of Kahler-Einstein edge metrics on asymptotically log del Pezzo surfaces.
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.
Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.
problem Proving mirror symmetry for specific Calabi-Yau surfaces.
method Adapting Hein's work, constructing asymptotically semi-flat Calabi-Yau metrics, and defining a mirror map.
result Existence and uniqueness of Calabi-Yau metrics on Y∖D. Study Moishezon twistor spaces using quartic hypersurfaces and Del Pezzo fibrations.
problem Classify Moishezon twistor spaces with specific half-anti-canonical systems.
method Utilize pluri-half-anti-canonical maps and quartic hypersurfaces to investigate twistor spaces.
result Complete classification of Moishezon twistor spaces with half-anti-canonical systems as pencils.
The aim of this paper is to construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of S2×S3. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler--Einstein metric on the Del Pezzo orbifold is then lifted to an Ei…
The study explores Sasaki-Einstein manifolds and their connections to field theories and AdS/CFT.
problem Understanding the geometric and algebraic properties of Sasaki-Einstein manifolds and their dual field theories.
method Algebraic and geometric methods to analyze Sasaki-Einstein manifolds and associated singularities.
result Proposed a conjecture to simplify the check of K stability for finding more SE metrics.
Extends BCOV invariant to pairs of Calabi-Yau manifolds and del Pezzo surfaces.
problem Constructing an invariant for Calabi-Yau manifolds and their pairs.
method Extending BCOV invariant to pairs (X,Y) where X is a compact Kaehler manifold and Y is a line bundle. result The extended BCOV invariant is equivalent to Yoshikawa's invariant for rigid del Pezzo surfaces and well-behaved under blow-up for m=1. The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.
problem Which finite subgroups of the mapping class group of a del Pezzo surface lift to the diffeomorphism group?
method Classification and partial answers for d≥7, equivariant connected sum for d=6. result Complete classification for d≥7, partial answer for d=6. For every smooth del Pezzo surface S, smooth curve C∈∣−KS∣ and β∈(0,1], we compute the α-invariant of Tian α(S,(1−β)C) and prove the existence of Kähler--Einstein metrics on S with edge singularities along C of angle 2πβ for β in certain interval. In particular we give lower bounds for the inva…
Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.
problem Investigating K-stability of specific del Pezzo surfaces.
method Relating K-stability to GIT stability of binary forms, proving stability and non-stability conditions.
result K-polystability and non-K-stability of quasi-smooth hypersurfaces.
Researchers describe a method to construct Ricci-flat metrics on a specific surface.
problem Constructing Ricci-flat metrics on a del Pezzo surface.
method Developed a framework and explicitly constructed the first-order deformation of the orthotoric metric.
result The deformation of the conformal Killing-Yano form does not exist.
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.
Researchers match complex affine structures in mirror constructions.
problem Matching complex affine structures in SYZ fibrations of Del Pezzo surfaces.
method Floer-theoretical gluing method to construct mirrors using immersed Lagrangians.
result The constructed mirror agrees with Carl-Pomperla-Siebert's mirror.