The perfect cone compactification of quotients of type IV domains

The perfect cone compactification is a toroidal compactification which can be defined for locally symmetric varieties. Let $$\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}$$ D L / O ~ + ( L ) ¯ p be the perfect cone compactification of the quotient of the type IV domain $$D_{L}$$ D L associated to an even lattice L. In our main theorem we show that the pair $${ (\overline{D_{L}/\widetilde{O}^{+}(L)}^{p}, \Delta /2) }$$ ( D L / O ~ + ( L ) ¯ p , Δ / 2 ) has klt singularities, where $$\Delta $$ Δ is the closure of the branch divisor of $${ D_{L}/\widetilde{O}^{+}(L) }$$ D L / O ~ + ( L ) . In particular this applies to the perfect cone compactification of the moduli space of 2d-polarised K3 surfaces with ADE singularities when d is square-free.

Double solid twistor spaces II: General case

In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied in [J. Differential Geom. 36 (1992), 451–491] and [Compos. Math. 82 (1992), 25–55] to the case of arbitrary signature. In particular, the branch divisor of the double covering is a cut of the rational threefold by a single quartic hypersurface. We determine a defining equation of the hypersurface in an explicit form. We also show that these twistor spaces interpolate LeBrun twistor spaces and the twistor spaces constructed in [J. Differential Geom. 82 (2009), 411–444].

Chamber Structure For Double Hurwitz Numbers

International audience Double Hurwitz numbers count covers of the sphere by genus $g$ curves with assigned ramification profiles over $0$ and $\infty$, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil (2005) have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein (2008) have determined the chamber structure and wall crossing formulas for $g=0$. We provide new proofs of these results, and extend them in several directions. Most importantly we prove wall crossing formulas for all genera. The main tool is the authors' previous work expressing double Hurwitz number as a sum over labelled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987). This approach to wall crossing appears novel, and may be of broader interest. This extended abstract is based on a new preprint by the authors. Les nombres de Hurwitz doubles dénombrent les revêtements de la sphère par une surface de genre $g$ avec ramifications prescrites en $0$ et $\infty$, et dont les autres valeurs critiques sont non dégénérées et fixées. Goulden, Jackson et Vakil (2005) ont prouvé que les nombres de Hurwitz doubles sont polynomiaux par morceaux en l'ordre des ramifications prescrites, et Shadrin, Shapiro et Vainshtein (2008) ont déterminé la structure des chambres et ont établis des formules pour traverser les murs en genre $0$. Nous proposons des nouvelles preuves de ces résultats, et les généralisons dans plusieurs directions. En particulier, nous prouvons des formules pour traverser les murs en tout genre. L'outil principal est le précédent travail des auteurs exprimant les nombres de Hurwitz doubles comme somme de graphes étiquetés. Nous identifions les étiquetages avec les points entiers à l'intérieur d'une chambre d'un arrangement d'hyperplans, qui sont connu pour donner une fonction polynomiale par morceaux. Notre étude des formules pour traverser les murs de ces fonctions se base sur un travail antérieur de Varchenko (1987). Cette approche paraît nouvelle, et peut être d'un large intérêt. Ce résumé élargi se base sur un papier nouveau des auteurs.

ON THE FUNDAMENTAL GROUP OF AN ABELIAN COVER

Let X, Y be smooth complex projective varieties of dimension n≥2 and let f: Y→X be a totally ramified abelian cover. Assume that the components of the branch divisor of f are ample. Then the map f*: π1(Y)→π1(X) is surjective and gives rise to a central extension: [Formula: see text] where K is a finite group. Here we show how the kernel K and the cohomology class c(f) ∈ H2(π1(X), K) of (1) can be computed in terms of the Chern classes of the components of the branch divisor of f and of the eigensheaves of [Formula: see text] under the action of the Galois group. Using this result, for any integer m>0, we construct m varieties X1,…, Xm no two of which are homeomorphic, even though they have the same numerical invariants and they are realized as covers of the same projective variety X with the same Galois group, branch locus and inertia subgroups.