Shapes of polyhedra and triangulations of the sphere

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm

Germs of de Rham cohomology classes which vanish at the generic point

We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding \'etale cohomology sheaves do fulfill Bloch-Ogus theory.Comment: latex 2e, 6 page

Supersolutions

We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and in superspace. Throughout we emphasize geometric aspects. The beginning chapters give a general discussion about supersymmetric field theories; then we move on to detailed computations of lagrangians, etc. in specific theories. An appendix details our sign conventions. This text will appear in a two-volume work "Quantum Fields and Strings: A Course for Mathematicians" to be published soon by the American Mathematical Society. Some of the cross-references may be found at http://www.math.ias.edu/~drm/QFT/Comment: 130 pages, AMSTe

Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involving a certain decomposition of the small diagonal in $S^3$ also proved by Beauville and the author}. We prove an analogue of such a decomposition of the small diagonal in $X^3$ for Calabi-Yau hypersurfaces $X$ in $\mathbb{P}^n$, which in turn provides strong restrictions on their Chow ring.Comment: Final version, to appear in Geometry \& Topolog

The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three

We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3

On the Deligne--Beilinson cohomology sheaves

We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is `effective' for $q=2$; in this case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge structures defined by singular cohomology, we are able to give a cohomological characterization of the Albanese kernel for surfaces with $p_g=0$.Comment: 12 pages, LaTeX 2.0

The Hodge--Poincar\'e polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \neq X_{(0)}^{s}\neq \emptyset$. We give formulae for the Hodge--Poincar\'e series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincar\'e polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.Comment: Final published version. arXiv admin note: text overlap with arXiv:math/0305346, arXiv:math/0305347 by other author

Product formula for p-adic epsilon factors

Let X be a smooth proper curve over a finite field of characteristic p. We prove a product formula for p-adic epsilon factors of arithmetic D-modules on X. In particular we deduce the analogous formula for overconvergent F-isocrystals, which was conjectured previously. The p-adic product formula is the equivalent in rigid cohomology of the Deligne-Laumon formula for epsilon factors in l-adic \'etale cohomology (for a prime l different from p). One of the main tools in the proof of this p-adic formula is a theorem of regular stationary phase for arithmetic D-modules that we prove by microlocal techniques.Comment: Revised version: some proofs and constructions detailed, notation improved, index of notation added ; 88 page