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arXiv1995

Spectral Covers

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

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arXiv1995

Algebraic Barth-Lefschetz theorems

Using results of Hironaka-Matsumura and Faltings, we prove a strong version of the well known Fulton-Hansen connectivity theorem for weighted projective spaces. As a consequence we get the following result. If $Y$ is an irreducible subvariety of the $n$-dimensional projective space (over a field of arbitrary characteristic), then the diagonal embedding $Δ_Y$ is $G_3$ in $Y\times Y$. This fact implies a generalized version (with a characteristic-free proof) of a result of Ogus (in char. zero) and Speiser (in positive characteristic).

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arXiv1995

The Coupled Seiberg-Witten Equations, vortices, and Moduli spaces of stable pairs

We introduce coupled Seiberg-Witten equations, and we prove, using a generalized vortex equation, that, for Kaehler surfaces, the moduli space of solutions of these equations can be identified with a moduli space of holomorphic stable pairs. In the rank 1 case, one recovers Witten's result identifying the space of irreducible monopoles with a moduli space of divisors. As application, we give a short proof of the fact that a rational surface cannot be diffeomorphic to a minimal surface of general type.

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arXiv1995

Stratified Picard--Lefschetz theory

The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the homology groups of the transversal slices of the corresponding strata.

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arXiv2014

Iitaka-Severi's Conjecture for Complex Threefolds

We prove the following generalization of Severi's Theorem: Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a dominant rational map $f$ from $X$ to Y$.

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arXiv1995

Lang's conjectures, Conjecture H, and uniformity

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. Let X be a variety of general type defined over a number field K. It was conjectured by S. Lang that the set of rational points X(K) is not Zariski dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is shown that the above conjecture of Lang implies the existence of a uniform bound on the number of K-rational points of all curves of fixed genus g over K. The paper [CHarM] has immediately created a chasm among arithmetic geometers. This chasm, which often runs right in the middle of the personalities involved, divides between loyal believers of Lang's conjecture, who marvel in this powerful implication, and the disbelievers, who try (so far in vain) to use this implication to derive counterexamples to the conjecture. In this paper we will attempt to deepen this chasm, using the techniques of [CHarM] and continuing [aleph], by proving more implications, some of which very strong, of various conjectures of Lang. Along the way we will often use a conjecture donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about Higher dimensional varieties, which is regarded very plausible among experts of higher dimensional algebraic geometry. In particular, we will show

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arXiv1995

Topological invariants of a fan associated to a toric variety

Associated to a toric variety $X$ of dimension $r$ over a field $k$ is a fan $Δ$ on $\Bbb R^r$. The fan $Δ$ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on $X$. The fan $Δ$ inherits the Zariski topology from $X$. In this article some cohomological invariants of $X$ are studied in terms of whether or not they depend only on $Δ$ and not $k$. Secondly some numerical invariants of $X$ are studied in terms of whether or not they are topological invariants of the fan $Δ$. That is, whether or not they depend only on the finite topological space defined on $Δ$. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the étale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicial.

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arXiv1995

Del Pezzo surfaces over Dedekind schemes

Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a smooth complex curve, and a conjectural answer if X is a cubic Del Pezzo surface over (nearly) arbitrary S.

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arXiv1995

Composite Differentiable Functions

We introduce a new point of view towards Glaeser's theorem on composite $C^\infty$ functions [Ann. of Math. 1963], with respect to which we can formulate a ``$C^k$ composite function property" that is satisfied by all semiproper real analytic mappings. As a consequence, we see that a closed subanalytic set $X$ satisfies the $C^\infty$ composite function property if and only if the ring $C^\infty (X)$ of $C^\infty$ functions on $X$ is the intersection of all finite differentiability classes.

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arXiv1995

Stability of the Poincaré bundle

Let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the complex numbers, and let $M_ξ$ denote the moduli space of stable bundles of rank $n$ and determinant $ξ$ on $C$, where $ξ$ is a line bundle of degree $d$ on $C$ and $n$ and $d$ are coprime. It is shown that the universal bundle $\cu_ξ$ on $C\times M_ξ$ is stable with respect to any polarisation on $C\times M_ξ$. It is shown further that the connected component of the moduli space of $\cu_ξ$ containing $\cu_ξ$ is isomorphic to the Jacobian of $C$.

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arXiv1995

Higher cohomology triples and holomorphic extensions

We introduce equations for special metrics, and notions of stability for some new types of augmented holomorphic bundles. These new examples include holomorphic extensions, and in this case we prove a Hitchin-Kobayashi correspondence between a certain deformation of the Hermitian-Einstein equations and our definition of stability for an extension.

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arXiv1995

Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces

We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for computing the Donaldson invariants of all rational surfaces in almost all chambers. In particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the projective plane of degree at most 50.

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arXiv1995

A Vector Bundle of Rank 2 on P1 x P3

The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric information. It is defined via the Serre construction by an abelian surface in P1 x P3. The bundle is stable with respect to O(1,1), its jumping lines are determined. Morover it is shown that it is closely related to a double structue on P1 x P1 embedded in P1 x P3 as well as to a certain pencil of quadrics of degree 4 in P3.

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arXiv1995

Residues in Toric Varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y-{0})/C* such that the toric residue becomes the local residue at 0 in Y.

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arXiv1995

Automorphisms of crepant resolutions for quotient spaces

A formula for calculating the Lefschetz number of an automorphism acting on a crepant resolution for a quotient of a Kahler manifold derived from an equivariant version of McKay correspondence. The latter is proven in some cases. As an application the Lefschetz numbers of of involutions acting on Calabi-Yau threefolds and their mirrors are compared.

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