arXiv subjects

Donu Arapura

Publications and source records attributed to Donu Arapura.

58 records · Page 4Linked to original sources

arXiv1997

Geometry of cohomology support loci II: integrability of Hitchin's map

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map to Sigma is an abelian variety. This should be interpreted as a nonabelian version of a theorem of Green and Lazarsfeld. The Hitchin map in this setting is in fact Simpson's generalization of it. The key point is to show that the general fibers of this map are lagrangian (where the target of the map is taken to be the image). This hinges on the fact that this property is essentially hereditary for hyperkaehler submanifolds. We establish various other properties of these sets including a codimension estimate which may be viewed as a generic vanishing theorem.

alg-geom↗

arXiv1996

Geometry of cohomology support loci for local systems I

Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $Σ^k(X)$ of one dimensional local systems on X with nonvanishing kth cohomology. We show that under certain conditions (X compact, X has a smooth compactification with trivial first Betti number, or k=1) $Σ^k(X)$ is a union of translates of sets of the form $f^*H^1(T,C^*)$, where $f:X \to T$ is a holomorphic map to a complex Lie group which is an extension of a compact complex torus by a product of C^*'s (these correspond to semiabelian varieties in the algebraic category). This generalizes earlier work of Beauville, Green, Lazarsfeld, Simpson and the author in the compact case. The main novelty lies in the proofs which involve consideration of Higgs fields with logarithmic poles. While a completely satifactory theory of such objects is still lacking, we are able to work out what we need in the rank one case by borrowing ideas from mixed Hodge theory. This will appear in the Journal of Algebraic Geometry.

alg-geom↗

arXiv1996

Intermediate Jacobians of moduli spaces

Let $SU_X(n,L)$ be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g curve X. Let $SU_X^s(n,L)$ denote the open subset parametrizing stable bundles. We show that if g>3 and n > 1, then the mixed Hodge structure on $H^3(SU_X^s(n, L))$ is pure of type ${(1,2),(2,1)}$ and it carries a natural polarization such that the associated polarized intermediate Jacobian is isomorphic J(X). This is new when deg L and n are not coprime. As a corollary, we obtain a Torelli theorem that says roughly that $SU_X^s(n,L)$ (or $SU_X(n,L)$) determines X. This complements or refines earlier results of Balaji, Kouvidakis-Pantev, Mumford-Newstead, Narasimhan-Ramanan, and Tyurin.

alg-geom↗

arXiv1992

Higgs line bundles, Green-Lazarsfeld sets,and maps of Kähler manifolds to curves

Let $X$ be a compact Kähler manifold. The set $\cha(X)$ of one-dimensional complex valued characters of the fundamental group of $X$ forms an algebraic group. Consider the subset of $\cha(X)$ consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree $d$. This set is shown to be a union of finitely many components that are translates of algebraic subgroups of $\cha(X)$. When the degree $d$ equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is shown that the number of equivalence classes (under a natural equivalence relation) of holomorphic maps, with connected fibers, of $X$ onto smooth curves of a fixed genus $>1$ is a topological invariant of $X$. In fact it depends only on the fundamental group of $X$.

math.AG↗