arXiv research

We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rat…

Survey on rational curves on complex surfaces, highlighting different approaches.

Method to create rational Seifert surfaces for knots in Lens space.

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.

Nonorientable surfaces in rational 4-manifolds linked to Pontrjagin square.

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are a…

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a ration…

The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…

Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.

The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.

A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or quadratic differential. Veech surfaces are a special class of translation surfaces …

K3 surfaces get a rational curve when a divisor is big and positive enough.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…

This paper completes the classification of certain surface singularities with rational homology disk smoothings.

Generalizes Lefschetz fibrations with rational homology disk smoothings.

Positive holomorphic sectional curvature on rational surfaces is characterized by Kähler metrics.

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Study trisections on rational elliptic surfaces to find new Zariski pairs.

The delta invariant of curves on rational surfaces is calculated using embedded topological and analytic methods.

Complete conjecture on rational curves on K3 surfaces.

The study connects periodic surface homeomorphisms to contact structures using rational open books.

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.

The paper studies cyclic covers of rational surfaces and their Hodge structures.

We study the change of moduli spaces of Gieseker-semistable torsion free rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the surfaces are rational with an effective anti-canonical divisor, the moduli spaces are linked by a series of flips (blowups and blowdowns). Using these results, we comput…

Study delta invariant of minimal generic curves on rational surfaces.

The study of symplectic fillings for rational cuspidal curves.

Classifies periodic points on regular and double n-gon surfaces.

New Lie group approach for envelope surface computation.

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …

Study delta invariant of curves on rational surfaces using topological methods.

Study contact structures on lens spaces, classifying rational knots.

Study homology groups of mapping and Torelli groups for surfaces with abelian covers.

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces…

Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.

New proof shows rationality of scl for non-filling curves.

We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.

Homology of surface coverings solved for genus 3 and above.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{\mathbf CP}^2}$ admits an exotic s…

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -χ(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constru…

Study counts specific surfaces in Montesinos knots with 4 rational tangles.

In this paper we study the topology of the space $\I_ω$ of complex structures compatible with a fixed symplectic form $ω$, using the framework of Donaldson. By comparing our analysis of the space $\I_ω$ with results of McDuff on the space $\cat J_ω$ of compatible almost complex structures on rational ruled surfaces, we…

Planar multilinks prove rational singularities in surface geometry.

The quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for complex rational and Enriques surfaces $X$ defined over $\R$ are shown to be diffeomorphic to connected sums of $\barCP2$, whenever $Y$ are simply connected.