Let R be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power seriesthere is defined a series (A(t)) [M ] , also with coefficients from R, so that all the usual properties of the exponential function hold. In the particular case A(t) = (1 − t) −1 , the series (A(t)) [M ] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.

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The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function Z DL (h, T ) of a quasi-ordinary power series h of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent Z DL (h, T ) = P (T )/Q(T ) such that almost all the candidate poles given by Q(T ) are poles. Anyway, these candidate poles give eigenvalues of the monodromy action of the complex of nearby cycles on h −1 (0). In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if h is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

ABSTRACT. -In this work we give a formula for the local Denef-Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef-Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler-Poincaré characteristic is three.

BACKGROUND AND PURPOSEPharmacological preconditioning (PPC) with mitochondrial ATP-sensitive K + (mitoKATP) channel openers such as diazoxide, leads to cardioprotection against ischaemia. However, effects on Ca 2+ homeostasis during PPC, particularly changes in Ca 2+ channel activity, are poorly understood. We investigated the effects of PPC on cardiac L-type Ca 2+ channels. EXPERIMENTAL APPROACHPPC was induced in isolated hearts and enzymatically dissociated cardiomyocytes from adult rats by preincubation with diazoxide. We measured reactive oxygen species (ROS) production and Ca 2+ signals associated with action potentials using fluorescent probes, and L-type currents using a whole-cell patch-clamp technique. Levels of the a1c subunit of L-type channels in the cellular membrane were measured by Western blot. KEY RESULTSPPC was accompanied by a 50% reduction in a1c subunit levels, and by a reversible fall in L-type current amplitude and Ca 2+ transients. These effects were prevented by the ROS scavenger N-acetyl-L-cysteine (NAC), or by the mitoKATP channel blocker 5-hydroxydecanoate (5-HD). PPC signficantly reduced infarct size, an effect blocked by NAC and 5-HD. Nifedipine also conferred protection against infarction when applied during the reperfusion period. Downregulation of the a1c subunit and Ca 2+ channel function were prevented in part by the protease inhibitor leupeptin. CONCLUSIONS AND IMPLICATIONSPPC downregulated the a1c subunit, possibly through ROS. Downregulation involved increased degradation of the Ca 2+ channel, which in turn reduced Ca 2+ influx, which may attenuate Ca 2+ overload during reperfusion. Abbreviations