Related papers: Nakai conjectures for isolated homogeneous hypersu…

Let $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the…

We show that there is a smooth complex projective variety, of any dimension greater than or equal to two, whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are…

The conjecture called algebraic Montgomery-Yang problem is still open for rational $\mathbb{Q}$-homology projective planes with cyclic quotient singularities having ample canonical divisor. All known such surfaces have a special birational…

Algebraic $K$-theory is a homology theory that behaves very well on sufficiently nice objects such as stable $C^*$-algebras or smooth algebraic varieties, and very badly in singular situations. This survey explains how to exploit this to…

We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y…

In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a…

A version of Kontsevich Formality theorem is proven for smooth DG algebras. As an application of this, it is proven that any quasiclassical datum of noncommutative unfolding of an isolated surface singularity can be quantized.

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras…

For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples. We show…

We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange…

The topology of the orbit space, $Y$, for the action of the complex conjugation on a complex surface, $X$, defined over reals, is studied. I give a criterion for blow-up stable triviality of $Y$ (which implies vanishing of its…

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism,…

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

We find and describe unexpected isomorphisms between two very different objects associated to hypersurface singularities. One object is the Milnor algebra of a function, while the other object associated to a singularity is the local ring…

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups.

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality…