We study the singularities of varieties obtained as infinitesimal quotients by $1$-foliations in positive characteristic. (1) We show that quotients by (log) canonical $1$-foliations preserve the (log) singularities of the MMP. (2) We prove that quotients by multiplicative derivations preserve many properties, amongst which most F-singularities. (3) We formulate a notion of families of $1$-foliations, and investigate the corresponding families of quotients.

We prove that the minimal exponent for local complete intersections satisfies an Inversion-of-Adjunction property. As a result, we also obtain the Inversion of Adjunction for higher Du Bois and higher rational singularities for local complete intersections.

Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder $\preccurlyeq$ on the set of constructible motivic functions, and an invariant $V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on $\preccurlyeq$ and $V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants $V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving $\preccurlyeq$, the metric entropy and our invariants $V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension $d$, relating $V_0$ (and $V_d$) and the motivic measure in dimension $d$.

Let $\mathcal {X}\to \mathbb {D}$ be a flat family of projective complex 3-folds over a disc $\mathbb {D}$ with smooth total space $\mathcal {X}$ and smooth general fibre $\mathcal {X}_t,$ and whose special fiber $\mathcal {X}_0$ has double normal crossing singularities, in particular, $\mathcal {X}_0=A\cup B$, with A, B smooth threefolds intersecting transversally along a smooth surface $R=A\cap B.$ In this paper, we first study the limit singularities of a $\delta $-nodal surface in the general fibre $S_t\subset \mathcal {X}_t$, when $S_t$ tends to the central fibre in such a way its $\delta $ nodes tend to distinct points in R. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on R along a $\delta $-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $\delta $-nodal surface in the general fibre of $\mathcal {X}\to \mathbb {D}$. As applications, we prove that there are regular irreducible components of the Severi variety of degree d surfaces with $\delta $ nodes in $\mathbb {P}^3$, for every $\delta \leqslant {d-1\choose 2}$ and of the Severi variety of complete intersection $\delta $-nodal surfaces of type $(d,h)$, with $d\geqslant h-1$ in $\mathbb {P}^4$, for every $\delta \leqslant {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$

In this paper, we prove that if a three-dimensional quasi-projective variety X over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section H of X. We also show that the same is true for klt singularities, which is a slight extension of [15]. In the course of the proof, we provide a sufficient condition for log canonical (resp. klt) surface singularities to be geometrically log canonical (resp. geometrically klt) over a field.

We consider the Bernstein–Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeros of the Bernstein–Sato polynomial. First, we prove each nonvanishing degree of the zeroth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein–Sato polynomial, generalizing a result of M. Saito (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeros of the Bernstein–Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted logarithmic comparison theorem, and more precise statements when f is additionally assumed to be homogeneous. Finally, when f defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeros of the Bernstein–Sato polynomial of f. We show all zeros except the candidate root $-2 + (2 / \deg(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only noncombinatorial candidate root $-2 + (2/ \deg(f))$ is in fact a zero of the Bernstein–Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\deg(f) - 1}$.

This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called “level” in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories.

This article describes local normal forms of functions in noncommuting variables, up to equivalence generated by isomorphism of noncommutative Jacobi algebras, extending singularity theory in the style of Arnold’s commutative local normal forms into the noncommutative realm. This generalisation unveils many new phenomena, including an ADE classification when the Jacobi ring has dimension zero and, by taking suitable limits, a further ADE classification in dimension one. These are natural generalisations of the simple singularities and those with infinite multiplicity in Arnold’s classification. We obtain normal forms away from some exceptional Type E cases. Remarkably, these normal forms have no continuous parameters, and the key new feature is that the noncommutative world affords larger families.

This theory has a range of immediate consequences to the birational geometry of 3-folds. The normal forms of dimension zero are the analytic classification of smooth 3-fold flops, and one outcome of NC singularity theory is the first list of all Type D flopping germs, generalising Reid’s famous pagoda classification of Type A, with variants covering Type E. The normal forms of dimension one have further applications to divisorial contractions to a curve. In addition, the general techniques also give strong evidence towards new contractibility criteria for rational curves.

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety $\operatorname {\mathrm {Pet}}_n$ and the opposite Schubert cell associated with the identity element $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is given by Ciocan–Fontanine and Givental–Kim. In this paper, we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties $\operatorname {\mathrm {Hess}}(N,h)$ and $\Omega _e^\circ $ in $\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. In other words, we generalize the Peterson’s statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan–Fontanine and Givental–Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety $\operatorname {\mathrm {Hess}}(N,h_m)$ and $\Omega _e^\circ $ is the intersection of certain Schubert variety and $\Omega _e^\circ $, where $h_m=(m,n,\ldots ,n)$ for $1<m<n$. We also see that $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $ is related with the cyclic quotient singularity.

We study the relation between the coregularity, the index of log Calabi–Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi–Yau pair $(X,B)$ of coregularity $1$ is at most $120\lambda ^2$, where $\lambda $ is the Weil index of $K_X+B$. This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano variety of absolute coregularity $0$ admits either a $1$-complement or a $2$-complement. In the case of Fano varieties of absolute coregularity $1$, we show that they admit an N-complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity $0$ admits either a $1$-complement or $2$-complement. Furthermore, a klt singularity of absolute coregularity $1$ admits an N-complement with N at most 6. This extends the classic classification of $A,D,E$-type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity $2$. In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least $3$, we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.

We analyze infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety X over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of the pairs given by a locally free sheaf and a subspace of its sections with a view toward Brill-Noether theory.

We realise Buchweitz and Flenner’s semiregularity map (and hence a fortiori Bloch’s semiregularity map) for a smooth variety X as the tangent of a generalised Abel–Jacobi map on the derived moduli stack of perfect complexes on X. The target of this map is an analogue of Deligne cohomology defined in terms of cyclic homology, and Goodwillie’s theorem on nilpotent ideals ensures that it has the desired tangent space (a truncated de Rham complex).

Immediate consequences are the semiregularity conjectures: that the semiregularity maps annihilate all obstructions, and that if X is deformed, semiregularity measures the failure of the Chern character to remain a Hodge class. This gives rise to reduced obstruction theories of the type featuring in the study of reduced Gromov–Witten and Pandharipande–Thomas invariants. We also give generalisations allowing X to be singular, and even a derived stack.

We generalize Illusie’s definition of the Atiyah class to complexes with quasi-coherent cohomology on arbitrary algebraic stacks. We show that this gives a global obstruction theory for moduli stacks of complexes in algebraic geometry without derived methods. We give a similar generalization of the reduced Atiyah class, and we show various useful properties for working with Atiyah classes, such as compatibilities between the reduced and ordinary Atiyah class, and compatibility with tensor products and determinants.

For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$-module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

Let R be a discrete valuation ring of field of fractions K and of residue field k of characteristic $p> 0$. In an earlier work, we studied the question of extending torsors over K-curves into torsors over R-regular models of the curves in the case when the structural K-group scheme of the torsor admits a finite flat model over R. In this paper, we first give a simpler description of the problem in the case where the curve is semistable using recent work in Holmes, Molcho, Orecchia, and Poiret (2023, Journal für die Reine und Angewandte Mathematik [Crelle’s Journal] 230, 115–159) and Molcho and Wise (2022, Compositio Mathematica 158, 1477–1562). Second, if R is assumed to be Henselian and Japanese, we solve the problem of extending torsors by combining our previous work together with results in Antei and Emsalem (2018, Nagoya Mathematical Journal 230, 18–34) and Phung and Dos Santos (2023, Algebraic Geometry 230, 1–40), including the case where the structural group does not admit a finite flat R-model.

The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

We study pencils of curves on a germ of complex reduced surface $(S,0)$. These are families of curves parametrized by $ \mathbb{P}^1 $ having 0 as the unique common point. We prove that for $w\in \mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$, where f and g are generators of the pencil.

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$, and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.

We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.