arXiv research

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

We prove that there are rational homology balls $B_p$ smoothly embedded in the $2$-handlebodies associated to certain knots. Furthermore we show that, if we rationally blow up the $2$-handlebody along the embedded rational homology ball $B_p$, then the resulting $4$-manifold cannot be obtained just by a sequence of ord…

Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres $C_n$ is replaced with a rational homology ball $B_n$, $n \geq 2$. Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic $C_n$ (given…

We prove a gluing formula for Seiberg--Witten invariants which describes in particular the behaviour of the invariant under blow-up and rational blow-down.

In this short note, we give an explicit construction of inequivalent Lefschetz pencils and fibrations of same genera on blow-ups of all rational and ruled surfaces. This complements our earlier results, concluding that every symplectic 4-manifold, after sufficiently many blow-ups, admits inequivalent Lefschetz pencils …

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

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

Let $X$ be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on $X$ are isotopic. This implies that blow-ups of these manifolds are unique, thus extending work of Biran. We also e…

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

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…

We construct potentially new manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ and $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ via rational blowdown surgery along certain $4$-valent plumbing graphs. This way all the graph classes from \cite{weighted} have a represen…

New forms generalize Whitney forms with rational coefficients for numerical analysis.

In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK…

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a Kähler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal m…

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply co…

Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as …

We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the …

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also con…

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillin…

We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who construct…

New proof shows unique symplectic fillings for certain surface singularity links.

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundl…

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

Nearby pinwheels are isotopic, solving Arnold's conjecture.

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

Formula derived for holomorphic Poisson blow-ups.

Study identifies numerical signs of blow-up in hydrodynamic equations.

Formula derived for Bott-Chern classes in complex blow-ups.

Explains blow-ups for Lie groupoids and algebroids, comparing different methods.

The paper derives a formula for Chow weights of toric blow-ups.

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

The paper examines the blow-up of Ricci curvatures in conformal metrics.

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …

The Yamabe flow can blow up in infinite time with small perturbations.

In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.

Study on spinor field equation on spheres, focusing on blow-up analysis.

New non-Kähler 3-folds constructed via log conifold transitions.

In this expository paper we review on the existence problem of Einstein-Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related resu…

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

Proves inextendibility of weak null singularities from curvature blow-up.