arXiv research

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.

Characterizes Wahl singularities in del Pezzo surface degenerations.

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…

A smooth curve found in a space of special surfaces.

We study global log canonical thresholds of del Pezzo surfaces.

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

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.

We prove functional identities for conic webs on del Pezzo surfaces.

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 $\mathbb{A}_{1}$, $\mathbb{A}_{2}$, $\mathbb{A}_{3}$, $\mathbb{A}_{4}$, $\mathbb{A}_{5}$ or $\mathbb{A}_{6…

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…

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.

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.

The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.

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.

We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.

Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.

Researchers prove mirror symmetry for certain non-compact Calabi-Yau surfaces.

The study classifies involutions on del Pezzo surfaces.

Researchers compute monodromy groups of surface families over quartic curves.

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is …

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.

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…

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 $(J, J^-_t, h_t)$ on $X$ by using $β$. In addition, if t…

Researchers solved a complex problem for a specific type of 4-manifolds.

Study monodromy factorizations for lines on del Pezzo surfaces.

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 $C^\infty$-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.

We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces X and vector fields v which are K-stable in the sense of Berman-Nystrom and therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kahler-Ricci soliton.

Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.

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.

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).

Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.

The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $β\in(0,1]$, 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…

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kahler-Einstein edge metrics on all but finitely…

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. 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 present new constructions of Kaehler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher dimensional examples are provided as well.

Researchers match complex affine structures in mirror constructions.

A five dimensional Sasaki-Einstein (SE) manifold provides a AdS/CFT pair for four dimensional $\mathcal{N}=1$ SCFT, and those pairs are very useful in studying field theory and AdS/CFT correspondence. The space of known SE manifolds is increased significantly in the last decade, and we initiated the study of various fi…

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an app…

Study of algebraic curves and surfaces in flag manifold using twistor geometry.

Solves Cheltsov-Rubinstein problem for complex surfaces with two boundary components.

We give an explicit counter-example to a conjecture of Kyusik Hong and Joonyeong Won about $α$-invariants of polarized smooth del Pezzo surfaces of degree one.

We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.

The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is not known in closed form. The manifold's toric structure reduces the Einstein equation to a single Monge-Ampere equation in two real dimensions. We numerically solve this nonlinear PDE using three different algorithms, and describe the result…