Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.     

Nuclear dimension, $$\mathcal{ }$$-stability,and algebraic simplicity for stably projectionless $$C^*$$C∗-algebras

Procesi bundles are certain vector bundles on symplectic resolutions of quotient singularities for wreath-products of the symmetric groups with the Kleinian groups. Roughly speaking, we can define Procesi bundles as bundles on resolutions that provide derived McKay equivalence. In this paper we classify Procesi bundles on resolutions obtained by Hamiltonian reduction and relate the Procesi bundles to the tautological bundles on the resolutions. Our proofs are based on deformation arguments and a connection of Procesi bundles with symplectic reflection algebras.     

Painlevé Classification Problems Featuring Essential Singularities

In this article we construct and solve all Painlevé-type differential equations of the second order and second degree that are built upon, in a natural well-defined sense, the "sn-log" equation of Painlevé, the general integral of which admits a movable essential singularity (elliptic function of a logarithm). This equation (which was studied by Painlevé in the years 1893–1902) is frequently cited in the modern literature to elucidate various aspects of Painlevé analysis and integrability of differential equations, especially the difficulty of detecting essential singularities by local singularity analysis of differential equations. Our definition of the Painlevé property permits movable essential singularities, provided there is no branching. While the essential singularity presents no serious technical problems, we do need to introduce new techniques for handling "exotic" Painlevé equations, which are Painlevé equations whose singular integrals admit movable branch points in the leading terms. We find that the corresponding full class of Painlevé-type equations contains three, and only three, equations, which we denote SD-326-I, SD-326-II, and SD-326-III, each solvable in terms of elliptic functions. The first is Painlevé's own generalization of his sn-log equation. The second and third are new, the third being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painlevé classification problems, including full generalizations of two other second-degree equations discovered by Painlevé, additional examples of exotic Painlevé equations and Painlevé equations admitting movable essential singularities, and third-order equations featuring sn-log and other essential singularities.     

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes, we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective \mathbb Q{\mathbb Q}-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev’s structure theorem for the cone of nef curves of projective terminal threefolds.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

On the dual of the mobile cone

We prove that the mobile cone and the cone of curves birationally movable in codimension 1 are dual to each other in the (K + B)-negative part for a klt pair (X, B). This duality of the cones gives a partial answer to the problem posed by Sam Payne. We also prove the cone theorem and the contraction theorem for the expanded cone of curves birationally movable in codimension 1.     

Sur l'anneau de cohomologie du schéma de Hilbert de C2

We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.     

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions

We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some L^p-type estimates, standard techniques of geometric P.D.E., and some ideas from symplectic geometry and calibration theory.     

Modular forms and special cubic fourfolds

We study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ADE singularities. In this paper, we show that the generating series of the degrees of such divisors is a level three modular form.     

Symplectic resolutions for nilpotent orbits (II)

Based on our previous work, Fu (Invent. Math. 151 (2003) 167–186), we prove that, given any two projective symplectic resolutions Z₁ and Z₂ of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z₁ is deformation equivalent to Z₂. This provides support for a ‘folklore’ conjecture on symplectic resolutions for symplectic singularities. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves

 We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon [9]. Received: 1 July 2002 / Revised version: 12 August 2002 Published online: 28 March 2003     

The geometry of the moduli space of one-dimensional sheaves

Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem.We also present the stable base locus decomposition of the space M6.As a byproduct,we obtain the Betti numbers of the moduli spaces,which confirm the prediction in physics.     

Monodromy conjecture for some surface singularities

In this work we give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three.     

Perfectly Weighted Graphs

 We present a classification of those graphs which arise as dual intersection graphs in the resolutions of complex surface singularities with perfect local fundamental group. Using computation intensive methods, we examine the possibilities for weights on the vertices corresponding to self-intersections of the exceptional curves. Received: August 19, 1998     

Resolving singularities with Cartan’s prolongation

A standard method for resolving a plane curve singularity is the method of blow-up. We describe a less-known alternative method which we call prolongation, in honor of Cartan’s work in this direction. This method is known to algebraic geometers as Nash blow-up. With each application of prolongation the dimension of the ambient space containing the new “prolonged” singularity increases by one. The new singularity is tangent to a canonical plane field on the ambient space. Our main result asserts that the two methods, blow-up and prolongation, yield the same resolution for unibranched singularities. The primary difficulties encountered are around understanding the prolongation analogues of the exceptional divisors from blow-up. These analogues are called critical curves. Most of the critical curves are abnormal extremals in the sense of optimal control theory as it applies to rank 2 distributions (2 controls). Dedicated to V. I. Arnol’d and his creative force     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

Equidistribution of divisors for sequences of holomorphic curves

We study holomorphic curves in ann-dimensional complex manifold on which a family of divisors parametrized by anm-dimensional compact complex manifold is given. If, for a given sequence of such curves, their areas (in the induced metric) monotonically tend to infinity, then for every divisor one can define adefect characterizing the deviation of the frequency at which this sequence intersects the divisor from the average frequency (over the set of all divisors). It turns out that, as well as in the classical multidimensional case, the set of divisors with positive defect is very rare. (We estimate how rare it is.) Moreover, the defect of almost all divisors belonging to a linear subsystem is equal to the mean value of the defect over the subsystem, and for all divisors in the subsystem (without any exception) the defect is not less than this mean value. This research was supported by RFBR grant No. 98-01-00867. Vladimir State Pedagogical University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 17–25, July–September, 2000. Translated by A. I. Shtern