arXiv research

Smooth approximation of integral cycles mod 2 in Riemannian manifolds.

The paper studies vanishing cycles in matrix singularities.

Smooth approximation of integral cycles in manifolds.

Study plane curve singularities to determine vanishing cycles and monodromy groups.

Introduces Lebesgue currents for real intersection theory.

The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.

An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriatel…

Constructs Morse homology for complex algebraic varieties.

New method identifies vanishing arcs for curve singularities.

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor …

Perfect pairing for tropical cycles on integral affine manifolds.

Study of complex surface singularities with generic analytic structure.

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by t…

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby …

New Stein fillings found for rational surface singularities.

An important class of contact 3--manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3--manifolds. As an application, we present a new obstruction for such singularities to admit rational homo…

Completed volumes match with combinatorial classes of the double ramification cycle.

Real Milnor fibres become contractible after attaching handles, matching classical results.

The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomol…

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…

In this paper we prove the infinitesimal uniqueness theorem for the Newton potential of non simply connected bodies using the singularity theory approach. We consider the Newtonian potentials of the domains in ${\bf R}^n$ boundaries of which are the vanishing cycles on the level hypersurface of a holomorphic function w…

Model for associative submanifolds in K3 fibrations.

Divides help construct fibered links from singularities.

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…

To each oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The obtained transformation L is the main object of study of the present paper. We pose a problem on the inversion of the transformation L. We shall show that this problem is closely rela…

The following numerical control over the topological equivalence is proved: two complex polynomials in $n\not= 3$ variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions $f_s \colon \mathbb{C}^n \to \mathbb{C}$ with isolated sin…

We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth surfaces aside from isolated points and therefore are J-holomorphic curves.

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of non-separating vanishing cycles and the number $D$ of singular fibers satisfy the inequality $N \…

Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.

Mathematical formulas for elliptic curve integrals solve anomaly equations.

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact struct…

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

It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set …

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is gene…

We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinat…

We investigate the bounded cohomology of Lefschetz fibrations. If a Lefschetz fibration has regular fiber of genus at least 2 and it has at least two distinct vanishing cycles, we show that its Euler class is not bounded. As a consequence, we exclude the existence of negatively curved metrics on Lefschetz fibrations wi…

We study the distribution of the common zero sets of $m$-tuples of holomorphic sections of powers of $m$ singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles c…

An R_2-move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations, flip and slip for example. In this paper, we first investigate how monodromies are chang…

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

The paper extends a theorem about momentum maps to singular symplectic spaces.

In this paper we will try to assess the multifractality displayed by the high-frequency returns of Madrid's Stock Exchange IBEX35 index. A Multifractal Detrended Fluctuation Analysis shows that this index has a wide singularity spectrum which is most likely caused by its long memory. Our findings also show that this lo…

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uni…

Constructs new coassociative fibrations for G2 manifolds.

This paper uses spectrum analysis to understand price behavior in the Indian stock market.

Study abelian factors in Lie algebras from graph edge labels.