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…

Study projective connections on surfaces using osculating spaces.

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

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

Classifies degenerations of complex projective plane with rational singularities.

Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.

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…

The paper constructs exotic complex projective surfaces using rational blowdowns.

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

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We…

Optimizes bounds for multiple T-singularities on surfaces.

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one i…

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber cla…

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is gi…

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group …

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motiva…

We present an approach of computing the intersection curve $\mathcal{C}$ of two rational parametric surface $§_1(u,s)$ and $§_2(v,t)$, one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve $G(v,t)=0$. By analyzing the topology …

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally conn…

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex projective surfaces. We prove that if the projective surface X is minimal, of g…

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives s…

The study provides a criterion for fractional-linear integrals of geodesics on surfaces.

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation the…

Study on rational projective planes with small index singularities.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

Survey on minimal rational curves and their geometric structures.

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical div…

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …

Study symplectic forms on manifolds to find Lagrangian pinwheels that can be separated.

In this short note, we present a construction of new symplectic 4-manifolds with non-negative signature using the complex surfaces on Bogomolov-Miyaoka-Yau line $c_1^2 = 9χ_h$, the fake projective planes and Cartwright-Steger surfaces. Our construction yields an infinite family of fake rational homology $(2n-1)\CP#(2n-…

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …

Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational…

All rational homology groups of unordered configuration spaces of the Moebius strip and the projective plane are calculated

Compact Kähler manifolds with positive curvature are projective and rationally connected.

Solves open problems on curved projective varieties.

Study ALE spaces via nodal curves and compactifications.

Classifies curves up to symplectic isotopy.

Study of rational curves in complex manifolds with specific normal bundles.

New method detects projective equivalences and symmetries in rational 3D curves.

Paper proves conditions for rational homology complex projective planes with singularities.

In this paper, we prove that if a compact Kähler manifold $X$ has a smooth Hermitian metric $ω$ such that $(T_X,ω)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a projective manifold $X$ is rationally connected, then the tautological line bundle $\mathscr{O}_{T…

Formula conjectured for rational cuspidal curves in projective plane.

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on…

The paper proves conditions for compact Kähler manifolds to be projective or rationally connected.

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $μ$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives man…

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). For each g …

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin numbers satisfying the Hirzebruch signature formula and a set of congruence relation…