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At least 217 records · Page 12Linked to original sources
arXiv2003

Horrocks theory and the Bernstein-Gelfand-Gelfand correspondence

We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, and establish a relation between HT-complexes and the Tate resolutions over the exterior algebra, which had been described by D. Eisenbud, G. Floystad, F.O. Schreyer in math.AG/0104203. The correspondence between HT-complexes and stable classes of vector bundles essentially translates into more fancy terms former results of Trautmann on representing Koszul complexes, which, in turn, were influenced by ideas of Horrocks. However, the relation between the Tate resolutions over the exterior algebra and HT-complexes might be new, although, perhaps, not a surprise to experts.

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arXiv2003

On a classical correspondence between K3 surfaces II

Let X be a K3 surface and H a primitive polarization of degree H^2=2a^2, a>1. The moduli space of sheaves over X with the isotropic Mukai vector (a,H,a) is again a K3 surface Y which is endowed by a natural nef element h with h^2=2. We give necessary and sufficient conditions in terms of Picard lattices N(X) and N(Y) when Y\cong X, generalising our results math.AG/0206158 for a=2. E.g. we show that Y\cong X if for one of α=\pm 1,\pm 2 which is coprime to a there exists h_1\in N(X) such that h_1^2= 2αa, H\cdot h_1\equiv 0\mod αa, and the primitive sublattice [H,h_1]_{pr} \subset N(X) contains x such that $x\cdot H=1$. We find all divisorial conditions on moduli of (X,H) (i.e for Picard number 2) which imply Y\cong X and H\cdot N(X)=Z. Some of these conditions were found in different form by A.N. Tyurin in 1987.

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arXiv2003

Mukai flops and derived categories II

This paper is a sequel to math.AG/0203287. A generalization of the Mukai flop has been studied by E. Markman. Here we call it a stratified Mukai flop. In this paper, we observe that, for a stratified Mukai flop: $X \to \bar{X} \leftarrow X^+$, the fiber product $X \times_{\bar{X}}X^+$ defines an isomorphism of Grothendieck groups $K(X) \to K(X^+)$, but, at the level of derived categories, the fiber product does not generally give an equivalence.

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arXiv2003

Nonemptiness of skew-symmetric degeneracy loci

Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a skew-symmetric bilinear form with values in a line bundle L and that Λ^2 V^* \otimes L is ample. Suppose that the maximum rank of the form at any point of X is r, where r>0 is even. The main result of this paper is that if d>2(N-r), then the locus of points where the rank of the form is at most r-2 is nonempty. This is a skew-symmetric analogue of the main result of math.AG/0305159; the proof is similar. If the hypothesis of ampleness is relaxed, we obtain a weaker estimate on the maximum dimension of X (and give a similar result for the symmetric case). We give applications to subschemes of skew-symmetric matrices, and to the stratification of the dual of a Lie algebra by orbit dimension.

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arXiv2003

Symplectic geometry of Frobenius structures

The purpose of the notes is to reiterate and expand the viewpoint, outlined in the paper math.AG/0110142 of T. Coates and the author, which recasts the concept of Frobenius manifold in terms of linear symplectic geometry and exposes the role of the twisted loop group L^{(2)}GL_N of hidden symmetries. New applications include a several line proof of the genus 0 Virasoro constraints and the quantum Hirzebruch-Riemann-Roch theorem in the theory of cobordism-valued Gromov-Witten invariants.

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arXiv2003

$\bar{partial}$-coherent sheaves over complex manifolds

This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\partial}$-coherent sheaves. We also include here the complete proof of our main Theorem.

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arXiv2003

Lines on contact Manifolds IIb

Let X be a complex-projective contact manifold whose second Betti-number is one. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding of X into a projective space whose image contains lines. Using methods introduced in math.AG/0206193, we show that X is covered by a compact family of rational curves, called "contact lines" that behave very much like the lines on the rational homogeneous examples: if x in X is a general point, then all contact lines through x are smooth, no two of them share a common tangent direction at x, and the union of all contact lines through x forms a cone over an irreducible, smooth base. As a corollary, we obtain that the tangent bundle of X is stable.

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arXiv2003

The Chow ring of the Cayley plane

We give a full description of the Chow ring of the complex Cayley plane, the simplest of the exceptional flag varieties. We describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions $Y_8$ introduced in our related preprint math.AG/0306328.

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arXiv2003

A weak Kawamata-Viehweg Vanishing Theorem

Using the techniques of J.-P. Demailly and Th. Peternell in math.AG/0208021 a weak version of Kawamata-Viehweg vanishing is proven for pseudo-effective line bundles on compact Kaehler manifolds. It is a weak version, since one has to adjust the line bundle $K_X + L$ with the upper regularized multiplier ideal sheaf $I_+(L)$ of $L$, and this is already not optimal in the nef case.

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arXiv2003

Smoothness of Equisingular Families of Curves

Francesco Severi showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface it thus makes sense to look for conditions which ensure that the family V of irreducible curves in the linear system |D| with precisely r singular points of types S1,...,Sr is T-smooth. Considering different surfaces including the projective plane, general surfaces in projective 3-space, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type gamma_a(S1)+...+gamma_a(Sr) < c*(D-K)^2, where gamma_a is some invariant of the singularity type and c is some constant. This generalises the results obtained by Greuel, Lossen and Shustin in math.AG/9903179 for the plane case, combining their methods and the method of Bogomolov instability, used by Chiantini and Sernesi in alg-geom/9602012. For many singularity types the $gamma_a$-invariant leads to essentially better conditions than the invariants previously used, and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.

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arXiv2005

Mirror Symmetry via Logarithmic Degeneration Data I

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. Then many statements about mirror symmetry which one hopes to be true for the general fibre should first be proved for this log CY space. In this paper we begin by discussing affine manifolds with singularities. Given such an affine manifold along with a polyhedral decomposition, we show how to construct a scheme consisting of a union of toric varieties. In certain non-degenerate cases, we can also construct log structures on these schemes. Conversely, given certain sorts of degenerations, one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration. Mirror symmetry is then obtained as a discrete Legendre transform on these affine manifolds, thus providing an algebro-geometrization of the Strominger-Yau-Zaslow conjecture. The deepest result of this paper shows an isomorphism between log complex moduli of a log CY space and log Kahler moduli of its mirror.

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arXiv2003

Random trees and moduli of curves

These are (not updated) notes from the lectures I gave in St.Petersburg in July of 2001. Their goal is to give an expository account of the proof of Kontsevich's combinatorial formula for intersections on moduli spaces of curves following math.AG/0101147.

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arXiv2004

"Brave New" Algebraic Geometry and global derived moduli spaces of ring spectra

We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topological modular forms as geometric S-stacks.

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arXiv2012

Generic vanishing, gaussian maps, and Fourier-Mukai transform

In the first part of this paper we prove a vanishing criterion for higher direct images of projective families of line bundles on a Cohen-Macaulay variety X. The result involves certain first-order deformations of certain curves on X, and makes essential use of the notion of global co-gaussian maps, a generalization of Wahl's gaussian maps. In the second part we apply the criterion above, combined with Fourier-Mukai transform on abelian varieties, to prove an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem. In fact we prove a stronger result concerning higher direct images of Poincaré line bundles, which -- in the compact Kähler setting -- was conjectured by Green and Lazarsfeld and was recently proved, by completely different methods, by Hacon (math.AG/0308198)

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arXiv2003

Gradients of odd theta functions

We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.

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arXiv2003

Coherent sheaves on generic compact tori

Let T be a compact complex torus, dim T>2. We show that the category of coherent sheaves on T is independent of the choice of the complex structure, if this complex structure is generic. The proof is independent of math.AG/0205210, where the same result was proven for K3 surfaces and even-dimensional tori.

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