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The aim of this paper is to construct an immersion of the Drinfeld moduli schemes in a finite product of infinite Grassmannians, such that they will be locally closed subschemes of these Grassmannians which represent a kind of flag varieties.
Let Q be a finite quiver without oriented cycles. Denote by U --> M the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext^i(U,U) = 0 for all i >0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra kQ of the quiver Q. If so, then the bounded derived category of finitely generated right kQ-modules is embedded into that of coherent sheaves on M.
In this paper we study Prym varieties and their moduli space using the well known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over the complex numbers; a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinite Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.
Let $G$ be a simple and simply connected complex Lie group. We discuss the moduli space of holomorphic semistable principal $G$ bundles over an elliptic curve $E$. In particular we give a new proof of a theorem of Looijenga and Bernshtein-Shvartsman, that the moduli space is a weighted projective space. The method of proof is to study the deformations of certain unstable bundles coming from special maximal parabolic subgroups of $G$. We also discuss the associated automorphism sheaves and universal bundles, as well as the relation between various universal bundles and spectral covers.
This manuscript from August 1995 (revised February 1996) studies the Kaehler cone of Calabi-Yau threefolds via symplectic methods. For instance, it is shown that if two Calabi-Yau threefolds are general in complex moduli and are symplectic deformations of each other, then their Kaehler cones are the same. The results are generalizations of those in the author's previous paper "The Kaehler cone on Calabi-Yau threefolds" (Inventiones math. 107 (1992), 561-583; Erratum: Inventiones math. 114 (1993), 231-233), where the behaviour of the Kaehler cone under deformations of the complex structure was studied -- these results may be recovered as a special case from this manuscript. The techniques used involve studying tamed almost complex deformations of the complex structure on the Calabi-Yau threefold, and in particular proving the non-vanishing of certain Gromov-Witten invariants, associated to codimension one faces of the Kaehler cone. This in turn involves a detailed study of primitive contractions on a smooth Calabi-Yau threefold, in particular for the case of Type III contractions. More detailed information concerning Gromov-Witten invariants associated to codimension one faces of the Kaehler cone may be found in a recent preprint of the author.
The aim of the paper is to give an `elementary' introduction to the theory of modules over operads and discuss three prominent examples of these objects - simplex, associahedron (= the Stasheff polyhedron) and cyclohedron (= the compactification of the space of configurations of points on the circle). Keywords: (right) module over an operad, module associated to a cyclic operad, Koszul module over an operad.
This is a detailed write-up of Kollár's course at the EMS summer school in Algebraic Geometry in Eger, Hungary, August 1996. The topics include definitions and examples of rational and unirational varieties, with special attention to varieties defined over non-algebraically closed fields, Segre's theorem on non-rationality of cubic surfaces of Picard number one, Manin's theorem that birationally equivalent cubic surfaces of Picard number one are projectively equivalent, and Kollár's method for constructing examples of low degree non-rational hypersurfaces. Includes many exercises and their solutions.
A contact structure on a complex manifold M is a corank 1 subbundle F of T(M) such that the bilinear form on F with values in the quotient line bundle L=T(M)/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample. If g is a simple Lie algebra, the unique closed orbit in P(g) (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry. In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M - - -> P(H0(M,L)*) associated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant.
We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of $L^2$-estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain $d''$-complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, $t$-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form $E=F \otimes L$, where $F$ is a $t$-nef vector bundle and $L$ is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.
The aim of this note is to give a simple definition of genus zero virtual orientation classes (or fundamental classes) for projective complete intersections or, more generally, for complete intersections in convex varieties, and to prove a push forward formula for them.
We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably self-contained, modulo the Special Adjoint Functor Theorem. An alternative approach, inspired by Neeman and based on recent results about "Brown Representability," is indicated as well. A section on applications and examples illustrates how these theorems synthesize a number of different duality-related results (local duality, formal duality, residue theorems, dualizing complexes...). A flat-base-change theorem for pseudo-proper maps leads in particular to sheafified versions of duality for bounded-below complexes with quasi-coherent homology. Thanks to Greenlees-May duality, the results take a specially nice form for proper maps and bounded-below complexes with coherent homology.
We define notions of direct and inverse limits in an $n$-category. We prove that the $n+1$-category $nCAT'$ of fibrant $n$-categories admits direct and inverse limits. At the end we speculate (without proofs) on some applications of the notion of limit, including homotopy fiber product and homotopy coproduct for $n$-categories, the notion of $n$-stack, representable functors, and finally on a somewhat different note, a notion of relative Malcev completion of the higher homotopy at a representation of the fundamental group.
We determine the cone of nef divisors on the Voronoi compactification A_g^* of the moduli space A_g of principally polarized abelian varieties of dimension g for genus g=2,3. As a corollary we obtain that the spaces A_g^*(n) with level-n structure are a minimal, resp. canonical, model for g=2, n>=4, resp. n>=5 and g=3, n>=3, resp. n>=4. We give two proofs: The easy and quick one reduces the problem to \bar M_g where we can use a result of Faber. This approach cannot be generalized to higher genus g. The main point of the paper is, therefore, to give a second proof using theta functions and a result of Weissauer. This technique can be at least partially generalized to higher genus. We formulate a conjecture for the nef cone of A_g^* for all g.
In this short note we give an explicit presentation of the algebra A_n generated by the curvature 2-forms of the standard Hermitiam line bundles over SL_n/B as the quotient of the polynomial ring. The difference between A_n and H^*(SL_n/B) reflects the fact that SL_n/B is not a symmetric space. Possible applications of A_n lie in the field of arithmetic intersection theory on flag varieties.
In this note we show the existence of a family of elliptic conic bundles in P^4 of degree 8. This family has been overlooked and in fact falsely ruled out in a series of classification papers. Our surfaces provide a counterexample to a conjecture of Ellingsrud and Peskine. According to this conjecture there should be no irregular m-ruled surface in P^4 for m at least 2.
In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (in a future paper).
This is my talk on the Bourbaki seminar, November 1996. It contains an elementary introduction to Borcherds' product formulas.
We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H. As an application, we derive recursive formulas for the number of such curves when the number is finite. These recursive formulas require as ``seed data'' only one input: there is one line in P^1 through two points. These numbers can be seen as top intersection products of various cycles on the Hilbert scheme of degree d rational or elliptic curves in P^n, or on certain components of $\mbar_0(P^n,d)$ or $\mbar_1(P^n,d)$, and as such give information about the Chow ring (and hence the topology) of these objects. The formula can also be interpreted as an equality in the Chow ring (not necessarily at the top level) of the appropriate Hilbert scheme or space of stable maps. In particular, this gives an algorithm for counting rational and elliptic curves in P^n intersecting various fixed general linear spaces. (The genus 0 numbers were found earlier by Kontsevich-Manin, and the genus 1 numbers were found for n=2 by Ran and Caporaso-Harris, and independently by Getzler for n=3.)