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At least 73 records · Page 4Linked to original sources
arXiv2005

Topology, geometry, and equations of normal surface singularities

This expository talk is an expanded version of a lecture at G.-M. Greuel's 60th Birthday Conference in Kaiserslautern in October, 2004. We survey recent work of Neumann-Wahl and others on the relation between topology and geometry of normal surface singularities, especially how to write explicit equations of a singularity with given topology, in case the link is a rational homology sphere (i.e., the resolution graph is a tree of rational curves). Specifically, under some reasonable conditions on the link, there is an explicit "complete intersection of splice type" whose link is the universal abelian cover of the original one, and for which the action of the covering group can be explicitly described. Recent work of T. Okuma shows every rational surface singularity can be described in this way. Full details are found in papers in Geometry and Topology, math.AG/0301165 and math.AG/0407287.

math.AG↗
arXiv2005

Orbifold Cohomology of ADE-singularities

We study Ruan's "cohomological crepant resolution conjecture" (see math.AG/0108195) for orbifolds with transversal ADE singularities. Let [Y] be such an orbifold, Y its coarse moduli space and Z the crepant resolution of Y. Following Ruan [math.AG/0108195], we have a deformation of the cohomology ring H^*(Z) using some Gromov-Witten invariants of Z. The resulting family of rings will be denoted by H^*(Z)(q_1,...,q_n), where q_1,...,q_n are complex parameters. In the A_n case we compute both the orbifold cohomology ring H^*_{orb}([Y]) and the family H^*(Z)(q_1,...,q_n). The former is achieved in general, the later up to an explicit conjecture on the Gromov-Witten invariants which is verified under additional, technical assumptions. We construct an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(-1) in the A_1 case, verifying Ruan's conjecture. In the A_2 case, the family H^*(q_1,q_2) is not defined for q_1=q_2=-1, so the conjecture should be slightly modified. However we give an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(q_1,q_2) with q_1=q_2 be a primitive third root of the unity, thus proving a slightly modified version of Ruan's conjecture. It is natural to conjecture that, in the A_n case, H^*_{orb}([Y]) is isomorphic to H^*(Z)(q_1,...,q_n) with q_1=...=q_n be a primitive (n+1)-th root of the unity.

math.AG↗
arXiv2005

Completions of $\C^*$-surfaces

Following an approach of Dolgachev, Pinkham and Demazure, we classified in math.AG/0210153 normal affine surfaces with hyperbolic $\C^{*}$-actions in terms of pairs of $\Q$-divisors $(D_+,D_-)$ on a smooth affine curve. In the present paper we show how to obtain from this description a natural equivariant completion of these $\C^*$-surfaces. Using elementary transformations we deduce also natural completions for which the boundary divisor is a standard graph in the sense of math.AG/0511063 and show in certain cases their uniqueness. This description is especially precise in the case of normal affine surfaces completable by a zigzag i.e., by a linear chain of smooth rational curves. As an application we classify all zigzags that appear as boundaries of smooth or normal $\C^*$-surfaces.

math.AG↗
arXiv2006

Constant families of t-structures on derived categories of coherent sheaves

We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of \cite{AJS}. As an application of the "constant" t-structures techniques we prove that every bounded nondegenerate t-structure on $D(X)$ with Noetherian heart is invariant under the action of a connected group of autoequivalences of $D(X)$. Also, we show that if $X$ is smooth then the only local t-structures on $D(X)$, i.e., those for which there exist compatible t-structures on $D(U)$ for all open $U\subset X$, are the perverse t-structures considered in math.AG/0005152.

math.AG↗
arXiv2006

Perverse sheaves on Artin stacks

In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in "The six operations for sheaves on Artin stacks I: Finite Coefficients" and "The six operations for sheaves on Artin stacks II: Adic Coefficients" (math.AG/0512097 and math.AG/0603680)

math.AG↗
arXiv2006

Correspondences of a K3 surface with itself via moduli of sheaves. I

Let $X$ be an algebraic K3 surface, $v=(r,H,s)$ a primitive isotropic Mukai vector on $X$ and $M_X(v)$ the moduli of sheaves over $X$ with $v$. Let $N(X)$ be Picard lattice of $X$. In math.AG/0309348 and math.AG/0606289, all divisors in moduli of $(X,H)$ (i. e. pairs $H\in N(X)$ with $\rk N(X)=2$) implying $M_X(v)\cong X$ were described. They give some Mukai's correspondences of $X$ with itself. Applying these results, we show that there exists $v$ and a codimension 2 submoduli in moduli of $(X,H)$ (i. e. a pair $H\in N(X)$ with $\rk N(X)=3$) implying $M_X(v)\cong X$, but this submoduli cannot be extended to a divisor in moduli with the same property. There are plenty of similar examples. We discuss the general problem of description of all similar submoduli and defined by them Mukai's correspondences of $X$ with itself and their compositions, trying to outline a possible general theory.

math.AG↗
arXiv2007

Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.

math.KT↗
arXiv2007

Multiplier ideals of sums via cellular resolutions

Fix nonzero ideal sheaves a_1,...,a_r on a normal Q-Gorenstein complex variety X. Fix any positive real number c, and consider the multiplier ideal J of the sum a_1+...+a_r with weighting coefficient c. We construct an exact sequence resolving J by sheaves over X that are direct sums of multiplier ideals for products a_1^{v_1}...a_r^{v_r} for various real vectors v such that v_1+...+v_r = c. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation T of the simplex of nonnegative real vectors summing to c. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on T, of a related monomial ideal. This acyclicity rests on a comparison between the homology of certain homology-manifolds-with-boundary and the homology of the simplicial complexes obtained by deleting collections of boundary faces from them. Our resolution implies the multiplier ideal sum formula J((a_1+...+a_r)^c) = \sum_{|v|=c} J(a_1^{v_1}...a_r^{v_r}), which implicitly follows from Takagi's proof of the two-summand formula (math.AG/0410612). We recover Howald's multiplier ideal formula for monomial ideals (math.AG/0003232) as a special case. Our resolution also yields a new exactness proof for the Skoda complex.

math.AG↗
arXiv2000

Torification and Factorization of Birational Maps

Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.

math.AG↗
arXiv1999

Stanley symmetric functions and quiver varieties

In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we discuss the relationship between this formula and Stanley symmetric functions. We show that the coefficients obtained when a Stanley symmetric function is expressed in the basis of Schur functions are special cases of a class of generalized Littlewood-Richardson coefficients appearing in the formula for quiver varieties. We also prove that Fulton's universal Schubert polynomials (math.AG/9702012) satisfy the usual rule for a Schubert polynomial of a product of two permutations.

math.CO↗
arXiv2005

On algebraic mixed Hodge substructures of H^2

This paper has been superseded to a great extent by the following: Paper: math.AG/0511558 Title: The Neron-Severi group of a proper seminormal complex variety Authors: L. Barbieri-Viale, A. Rosenschon & V. Srinivas Paper: math.AG/0102150 Title: Deligne's Conjecture on 1-Motives Authors: Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito Journal-ref: Ann. of Math. (2), Vol. 158 (2003), no. 2, 593--633

math.AG↗
arXiv2008

Constructible motivic functions and motivic integration

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration. These results have been announced in math.AG/0403349 and math.AG/0403350. Main results and statements unchanged. Many minor slips corrected and some details added.

math.AG↗
arXiv2000

Towards a combinatorial Intersection Cohomology for Fans

The real intersection cohomology of a toric variety is described in a purely combinatorial way using methods of elementary commutative algebra only. We define, for arbitrary fans, the notion of a ``minimal extension sheaf'' on the fan as an axiomatic characterization of the equivariant intersection cohomology sheaf. This provides a purely algebraic interpretation of Stanley's generalized f- and g-vector of an arbitrary polytope or complete fan under a natural vanishing condition. -- The results presented in this note originate from joint work with G.Barthel, J.-P.Brasselet and L.Kaup, continuing earlier research (see math.AG/9904159). A detailed exposition will appear elsewhere (see math.AG/0002181).

math.AG↗
arXiv2000

The first main theorem on complements: from global to local

The aim of this paper is to clarify and generalize techniques of works alg-geom/9711024 (see also math.AG/9810097 and math.AG/9901004). Roughly speaking, we prove that for local Fano contractions the existence of complements can be reduced to the existence of complements for lower dimensional projective Fano varieties.

math.AG↗
arXiv2002

On classification of Lorentzian Kac-Moody algebras

We discuss a general theory of Lorentzian Kac--Moody algebras which should be a hyperbolic analogy of the classical theories of finite-dimensional semi-simple and affine Kac-Moody algebras. First examples of Lorentzian Kac-Moody algebras were found by Borcherds. We consider general finiteness results about the set of Lorentzian Kac--Moody algebras and the problem of their classification. As an example, we give classification of Lorentzian Kac--Moody algebras of the rank three with the symmetry group which is an extended paramodular group. Perhaps, this is the first example when a large class of Lorentzian Kac--Moody algebras was classified. This paper is closely related with our papers math.AG/9810001, math.AG/0010329

math.QA↗
arXiv2008

Varieties with quadratic entry locus, II

We continue the study, begun by the second author in math.AG/0701889, of secant defective manifolds having "simple entry loci". We prove that such manifolds are rational and describe them in terms of tangential projections. Using also our results in math.AG/0701885, their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of rational homogeneous manifolds. We also provide evidence that the classification of linearly normal dual defective manifolds with Picard group generated by the hyperplane section should follow along the same lines.

math.AG↗
arXiv2011

Harmonic analysis and the Riemann-Roch theorem

This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from these papers and two-dimensional adelic theory we obtain the Riemann-Roch formula on a projective smooth algebraic surface over a finite field.

math.AG↗