arXiv research

The paper finds diffeomorphic complex intersections with distinct Hodge numbers.

Positivity of intersections in 4-manifolds leads to taming symplectic structures.

Condition for intersection of real flag manifolds in complex flag manifold.

Mapping class group subgroups yield quasi-isometric curve complex.

The study finds that most minimal surfaces in generic 4D manifolds intersect in complex ways.

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

We give the diffeomorphism classification of complete intersections with S^1-symmetry in dimension less than or equal to 6. In particular, we show that a 6-dimensional complete intersection admits a smooth non-trivial S^1-action if and only if it is diffeomorphic to the complex projective space or the quadric. We also …

3-manifold triangulation can be reconstructed from its intersection matrix.

Study the intersection of positive closed currents using tangent currents and King's residue formula.

Let $X$ be a contractible $2$-complex which is a union of two contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\cap Z$ contractible as well? In this note, we prove that the inclusion-induced map $π_{1}(Y\cap Z)\rightarrow π_{1}(Z)$ is injective if $Y$ is $π_{1}$-injective subcomplex in a locally CAT(0) 2-co…

New method to compute homology and intersection form of 4-manifolds.

We compute the alpha invariant of any smooth complex projective spin complete intersection of complex dimension $1 \; ({\rm mod} \; 4)$. We prove that the alpha invariant depends only on the total degree and Pontryagin classes. Our findings are consistent with a long-standing conjecture, often called the Sullivan Conje…

Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.

We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for $J$-holomorphic curves in almost complex $4$-manifolds to higher dimensions. As an applicat…

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are …

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at most $\ell$ on a combinatorial surface of complexity $n$ we describe simple algo…

The primitive cohomology of Calabi-Yau intersections is described using a twisted de Rham complex.

Super efficient geodesics have a unique vertex in the complex of curves.

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of…

An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We present an explicit description of the Chow rings of …

A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of two curves can be measured by the sum of all subsurface projection distances betw…

Proves a conjecture about Lagrangian intersections using new theory.

In this note, we prove that the Witten genus of nonsingular string complete intersections in product of complex projective spaces vanishes. Our result generalizes a known result of Landweber and Stong (cf. [HBJ]).

Probabilistic theory counts intersections in Riemannian spaces.

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

2D complexes can be almost-embedded in 4D space without self-intersections.

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the first author and further answering a question of Malestein-Rivin-Theran. To dist…

Establishing criteria for top cell inertness in complexes.

Study on symmetry defects of complete intersections in complex space.

The paper extends Busemann's inequalities to complex and quaternionic spaces.

Quotients $Y=X/conj$ by the complex conjugation $conj\: X\to X$ for complex surfaces $X$ defined over $\R$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums $\#_n CP^2\#_m\barCP^2$ if $w_2(Y)\ne0$, or into $\#_n(S^2\times S^2)$ if $w_2(Y)=0$. The author proves this prope…

Classifies certain graph 2-braid groups up to quasi-isometry.

Torsion sensitive intersection homology was introduced to unify several versions of Poincare duality for stratified spaces into a single theorem. This unified duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. In this paper we consider for…

Suppose a smooth planar curve $γ$ is $2π$-periodic in the $x$ direction and the length of one period is $\ell$. It is shown that if $γ$ self-intersects, then it has a segment of length $\ell- 2π$ on which it self-intersects and somewhere its curvature is at least $2π/(\ell - 2π)$. The proof involves the projection $Γ$ …

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…

Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of special lagrangian and lagrangian varieties in certain homogeneous spaces.

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs inter…

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} \[ <, >: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}. \] Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$…

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller space…

The paper studies geometric loci and their invariants in complex dynamics.

Let $P : Σ\rightarrow S$ be a finite degree covering map between surfaces. Rafi and Schleimer show that there is an induced quasi-isometric embedding $Π: \mathcal{C}(S) \rightarrow \mathcal{C}(Σ)$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}(Σ)$ using minimal intersection num…

Study on complexity of systolic geodesics on Bolza surface.

Continuous curves inscribe isosceles trapezoids in complex plane.

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfa…

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…