Given a map : X Y between F -analytic manifolds over a local field F of characteristic 0, we introduce an invariant () measuring the integrability of pushforwards of smooth compactly supported measures by , together with a local version (; x) near x X.These invariants are closedly tied to the singularities of .When Y is onedimensional, we provide an explicit formula for (; x) and show that it is asymptotically equivalent to classical singularity invariants, in particular the F -log-canonical threshold lct F ( -(x); x).In general, we prove that (; x) is bounded below by the F -log-canonical threshold = lct F (J ; x) of the Jacobian ideal J , with equality when dim Y = dim X.For polynomial maps : X Y between smooth algebraic Q-varieties, we show that the condition " ( F ) = over a large family of local fields" is equivalent to being flat with semi-log-canonical fibers.

We study the Kodaira dimension of moduli spaces of polarized hyperkhler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten-dimensional variety.This question was studied by Gritsenko-Hulek-Sankaran in the cases of K3 [2] -and OG10-type when the divisibility of the polarization is 1.We generalize their results to higher dimension and divisibility.As a main result, for almost all dimensions 2n, we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperkhler varieties of K3 [n] -type is of general type. K3[n] ,2d and M OG10,2d parametrize pairs (X, H), with X a projective hyperkhler variety of K3 [n] -type or OG10-type, respectively,

We study the Kodaira dimension of the compactified n-fold Kuga variety over the moduli space of principally polarised abelian g-folds.We construct a suitable compactification, which we call a Namikawa compactification, and show that in most cases it has canonical singularities.We then use results about the slope of Siegel modular forms to determine the Kodaira dimension for all g > 1 and n > 0.

A Brill-Noether degeneracy locus is the closure in Pic d (C) of the locus of line bundles with a specified rank function r(a, b) = h 0 (C, L(-ap -bq)).These loci generalize the classical Brill-Noether loci W r d (C) as well as Brill-Noether loci with imposed ramification.For general (C, p, q), we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about W r d (C).The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated with the rank function, or alternatively the number of saturated chains in the Bruhat order.The essential tool is a versality theorem for a certain pair of flags on Pic d (C), conjectured by Melody Chan and the author.

For a closed embedding of smooth schemes X S with a fixed first-order splitting, one can construct Hochschild-Kostant-Rosenberg (HKR) isomorphisms between the derived scheme X R S X and the total space of the shifted normal bundle N X/S [-1], thanks to Arinkin-Cldraru, Arinkin-Cldraru-Hablicsek, and Grivaux.In this paper, we study the functoriality property of the HKR isomorphisms for a sequence of closed embeddings X Y S. The HKR isomorphism is functorial when a certain cohomology class, which we call the Bass-Quillen class, vanishes.We obtain Lie-theoretic interpretations for the HKR isomorphisms and for the Bass-Quillen class as well.

We establish several foundational results regarding the Grothendieck-Springer affine fibration.More precisely, we prove some constructibility results on the affine Grothendieck-Springer sheaf and its coinvariants.We enrich it with a group of symmetries, analogously to the situation of Hitchin fibration.We prove some perversity statements once we take some derived coinvariants, and we construct some specialization morphisms for the homology of affine Springer fibers.

We show that the semistable locus is the unique maximal open substack of the moduli stack of principal bundles over a curve that admits a schematic moduli space.For rank 2 vector bundles, it coincides with the unique maximal open substack that admits a separated moduli space, but for higher rank there exist other open substacks that admit separated moduli spaces.

We compare tame actions in the category of schemes with torsors in the category of log schemes endowed with the log flat topology. We prove that actions underlying log flat torsors are tame. Conversely, starting from a tame cover of a regular scheme that is an fppf torsor on the complement of a divisor with normal crossings, it is possible to build a unique log flat torsor that dominates this cover. In brief, the theory of log flat torsors gives a canonical approach to the problem of extending torsors into tame covers.