Abstract Brasselet, the second author and Yokura introduced Hodge-theoretic Hirzebruch-type characteristic classes $$IT_{1, *}$$ I T 1 , ∗ , and conjectured that they are equal to the Goresky-MacPherson L -classes for pure-dimensional compact complex algebraic varieties. In this paper, we show that the framework of Gysin coherent characteristic classes of singular complex algebraic varieties developed by the first and third author in previous work applies to the characteristic classes $$IT_{1, *}$$ I T 1 , ∗ . In doing so, we prove the ambient version of the above conjecture for subvarieties in a Grassmannian. Since the homology of Schubert subvarieties injects into the homology of the ambient Grassmannian, this implies the conjecture for all Schubert varieties in a Grassmannian. We also study other algebraic characteristic classes such as Chern classes and Todd classes (or their variants for the intersection cohomology sheaves) within the framework of Gysin coherent characteristic classes.

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically étale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.

We study a form of dévissage for generation in derived categories of Noetherian schemes. First, we extend a result of Takahashi from the affine context to the global setting, showing that the bounded derived category is classically generated by a perfect complex together with structure sheaves of closed subschemes supported on the singular locus. Second, we make an observation for how generation behaves under the derived pushforward of a proper surjective morphism between Noetherian schemes. These results enable us to explicitly identify strong generators for projective schemes with isolated singularities and for singular varieties over a perfect field.

Abstract We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti‐canonical class. In the Appendix, we show that such an infinite family of smooth rationally elliptic 3‐folds does not exist.

Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of k -planes through the curve. These are singular varieties, with each secant variety being singular along the previous one. We study invariants of the singularities for these varieties. In the case of an arbitrary curve, we compute the intersection cohomology in terms of the cohomology of the curve. We then turn our attention to rational normal curves of even degree. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincaré duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.

Abstract We discuss the role of hyperelliptic fibrations in F-theory. For each even integer n we give a noncompact Calabi-Yau threefold X containing a hyperelliptically fibered surface Y, such that X and Y are homotopy equivalent and c 2(X) = n. We investigate two distinct cases depending on the position of the hyperelliptic fibration. First, we propose to extend F-theory considering hyperelliptic fibrations, giving an identification between the determinant of the period matrix and the axio-dilaton. Such an identification requires that the curve satisfies an appropriate criterium which we describe. Our explicit examples have split Jacobian, preserve the same number of degrees of freedom of usual F-theory while allowing for the appearance of a greater variety of singularities. Second, when the hyperelliptic fibration is contained in the base of a Calabi-Yau fourfold, we show that tadpole cancellation conditions are satisfied for arbitrarily large values of c 2(X).

We begin by presenting a version of the Poincaré–Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex K . To that end we construct a cellular sheaf on K whose cohomology with compact support is isomorphic to the homology of the initial cosheaf. Thereafter we use the first result to generalise the tropical version of the Lefschetz Hyperplane Section Theorem to some singular tropical toric varieties and singular tropical hypersurfaces.

We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group K with respect to the standard Lie–Poisson structure. These systems generalize key properties of Gelfand–Zeitlin systems: (A) the pullback to any Hamiltonian K -manifold defines a Hamiltonian torus action on an open dense subset, (B) if the K -manifold is multiplicity-free, then the resulting torus action is completely integrable, and (C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand–Zeitlin systems and Gelfand–Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh’s construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piecewise, has a flow whose limit exists and defines a continuous degeneration map.

We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.

We compute the Du Bois complexes of abstract cones over singular varieties, and use this to describe the local cohomological dimension and the non-positive K-groups of such cones.

Abstract We study Kähler-Einstein metrics on singular projective varieties. We show that under an approximation property with constant scalar curvature metrics, the metric completion of the smooth part is a noncollapsed RCD space, and is homeomorphic to the original variety.

Degenerate complex Monge-Ampère equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and $W^{3,p}$ estimate for a class of degenerate complex Monge-Ampère equations.

Abstract This paper delves into the study of curvilinear Hilbert schemes associated with a singular variety $(X,0)$ and the relationship between their motivic classes and the motivic measure on the arc scheme $X_\infty $ of $X$ introduced by Denef and Loeser. We introduce an Igusa zeta function specifically tailored for curvilinear Hilbert schemes for which we provide an explicit formulation in terms of an embedded resolution of the singularity, and we consequently obtain a recursive formula to compute the motivic classes of curvilinear Hilbert schemes in terms of the resolution. In addition, the paper explores and analyzes the geometry and combinatorics of curvilinear Hilbert schemes in the context of plane curve singularities and their topological invariants.

Abstract We study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have ‐Du Bois singularities (in a sense that was proposed in Shen, Venkatesh, and Vo [On k‐Du Bois and k‐rational singularities, arXiv e‐prints (June 2023), arXiv:2306.03977]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre‐1‐rational. From these results, we deduce several consequences, including a Kodaira–Akizuki–Nakano type vanishing result for the reflexive differential forms of the secant varieties.

Abstract In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.

The Morrison–Kawamata cone conjecture for singular symplectic varieties

Suppose that X is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic.Using ideas of Takamatsu-Yoshikawa and Bhatt et al., we define a notion of a + + +-test ideal on X, including for divisors and linear series.We obtain global generation results in this setting that generalize the well-known global generation results obtained via multiplier ideal sheaf techniques in characteristic zero and via test ideals in characteristic p > 0. We also obtain applications to the order of vanishing of linear series and to the diminished base locus in mixed characteristic similar to results of Ein-Lazarsfeld-Mustat -Nakamaye-Popa, Nakayama, and Mustat in the equal-characteristic case. IntroductionOne of the most useful properties of multiplier ideals J(X, ) on a projective variety in characteristic zero is Nadel vanishing (a variant of Kawamata-Viehweg vanishing).It states that H i (X, O X (K X + M ) J(X, )) = 0 for i > 0 assuming that K X + M is a Cartier divisor such that M - is ample [Nad89, Laz04].This in turn implies effective global generation results.For instance, on a nonsingular projective variety X of dimension n, we have thatis globally generated for m n when L is an ample divisor such that O X (L) is globally generated; see [Laz04, Proposition 9.4.26].Building on global generation results for O X (K X +mL) for mildly singular varieties in positive characteristic [Smi97, Kee08, Har01], similar results were also obtained for test ideals (X, ) in characteristic p > 0, where instead of using resolution of singularities and Nadel/Kawamata-Viehweg vanishing, one uses Frobenius and Serre/Fujita vanishing.Indeed, we have that if M and L are as above, then (X, ) O X (K X + M + mL)

Quasiordinary power series were introduced by Jung at the beginning of the 20th century, and were not paid much attention until the work of Lipman and, later on, Gao. They have been thoroughly studied since, as they form a very interesting family of singular varieties, whose properties (or at least many of them) can be encoded in a discrete set of integers, much as what happens with curves. Hironaka proposed a generalization of this concept, the so-called ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-quasiordinary power series, which has not been examined in the literature in such detailed way. This paper explores the behavior of these series under the resolution process in the surface case.

For the constraint variety in symplectic manifold, the solvable Hamiltonian vector fields on the constraint are investigated. According to P.A.M. Dirac [3], the space of solvable Hamiltonian systems is determined by the geometric restriction of the symplectic form to the constraint. Solvability condition of the generalized Hamiltonian systems is extended to singular varieties and applied under some assumption on singularities. The constraint being a smooth submanifold in a symplectic space was considered in [6]. In this paper, we investigate the solvability of generalized Hamiltonian systems and the constraint invariants on singular constraints in the constant rank case.

Abstract We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.