arXiv research

We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $β$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor …

We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor $Y$. Assuming the data in question is invariant under an $S^1$-action (locally around $Y$) we prove that this density function has a distri…

Study multiplicity of non-acyclic SL2-representations and L-functions of Whitehead links.

Study shows quantum behavior near infinity in metric asymptotics.

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differe…

Study shows hypercomplex twistor spaces lack divisors and special metrics.

Study b-divisors on Kähler manifolds linking them to currents.

Abstract: Study motion of divisors on curves with topological constraints.

Study contact geometry of symplectic divisors, invariant under specific transformations.

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

For arbitrary quantizable compact Kaehler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realised via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vec…

New stability criteria for Fano varieties using generalized b-divisors.

In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

The paper proves extension theorems for holomorphic sections from divisors.

Study Lagrangian Floer theory in smooth divisor complements.

This paper studies Poisson structures defined by divisor ideals.

Study locates divisors in Hodge bundle with specific properties.

The paper provides a combinatorial criterion for realizing tropical pluri-canonical divisors.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …

Study shows normal distribution in divisor counts of random sections on complex manifolds.

Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

New complete Calabi-Yau metrics found in complex space.

Study of zero-divisors in sedenions via determinant factorization.

For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus.

The paper proves finiteness for stable Lagrangian fibrations with a given divisor.

Injective homomorphism proves no zero divisors in Roger-Yang skein algebra.

The paper classifies and studies symplectic and contact properties of circular spherical divisors.

The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.

Abstract: Study of metrics on line bundles over complex varieties.

Introduces algorithm for Weinstein handlebodies of certain divisors.

Study Weinstein structures on toric divisors' complements.

Recently it was shown by H. Guenancia and M. Paun that a singular metric satisfying the conical Kahler-Einstein equation with a simple normal crossing divisor is equivalent to a conical metric along that divisor. In this note, we present an alternative proof of their theorem.

We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(Γ, D)$ consisting of a stable tropical curve $Γ$ …

Normal forms and moduli stacks for flat connections on complex manifolds.

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimens…

Study projective klt pairs with nef anti-canonical divisor and their properties.

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Au…

We classify simple singularities of functions on space curves. We show that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the bifurcations diagram of functions is true, and both this diagram and the discriminant a…

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which …

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve …

The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.

Proves functional equation for twisted Ruelle zeta function on hyperbolic surfaces.

Formulae count square-tiled surfaces in genus two.

In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at $E_1$, and we al…

We develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.

In this paper, Floer homology for Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor is discussed. The main new feature of this construction is that we do not make any assumption on positivity or negativity of the divisor. To achieve this goal, we use a compactification o…

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pair…