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Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (C n , 0) → (C , 0) . We introduce a new derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let I = (f , ∂ f ∂ x 1 , … , ∂ f ∂ x n ) and Δ (I) : = { g ∣ g , ∂ g ∂ x 1 , … , ∂ g ∂ x n ∈ I } , then A Δ (V) : = O n / Δ (I) and L Δ (V) : = Der (A Δ (V) , A Δ (V)) . Their dimensions as a complex vector space are denoted as β (V) and δ (V) , respectively. δ (V) is a new invariant of singularities. In this paper we study the new local algebra A Δ (V) and the derivation Lie algebra L Δ (V) , and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimation conjectures for β (V) and δ (V) when (V, 0) are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities. Lastly, we provide an application of β (V) and δ (V) to distinguishing contact classes of singularities. [ABSTRACT FROM AUTHOR]

Following Shokurov's idea, we give a simple proof of the ACC conjecture for minimal log discrepancies for surfaces. [ABSTRACT FROM AUTHOR]

We relate the Brasselet number of a complex analytic function-germ defined on a complex analytic set to the critical points of its real part on the regular locus of the link. Similarly we give a new characterization of the Euler obstruction in terms of the critical points on the regular part of the link of the projection on a generic real line. As a corollary, we obtain a new proof of the relation between the Euler obstruction and the Gauss–Bonnet measure, conjectured by Fu. [ABSTRACT FROM AUTHOR]

In this work, we show that Hölder equivalence of analytic functions germs (R 2 , 0) → (R , 0) admits continuous moduli. [ABSTRACT FROM AUTHOR]

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps f : (R n , 0) → (R k , 0) , where n ≥ k ≥ 2 , with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class C ℓ , ℓ ≥ 2 , with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class C ℓ between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations. [ABSTRACT FROM AUTHOR]

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the "Groebner fan" of such an ideal is actually a polyhedral fan and that a subvariety defined by a Newton non-degenerate ideal on a toric variety X σ admits a toric embedded resolution of singularities Z ⟶ X σ. [ABSTRACT FROM AUTHOR]

We provide a new, geometric proof of the motivic monodromy conjecture for non-degenerate hypersurfaces in dimension 3, which has been proven previously by the work of Lemahieu–Van Proeyen and Bories–Veys. More generally, given a non-degenerate complex polynomial f in any number of variables and a set B of B 1 -facets of the Newton polyhedron of f with consistent base directions, we construct a stack-theoretic embedded desingularization of f - 1 (0) above the origin, whose set of numerical data excludes any known candidate pole of the motivic zeta function of f at the origin that arises solely from facets in B . We anticipate that the constructions herein might inspire new insights as well as new possibilities towards a solution of the conjecture. [ABSTRACT FROM AUTHOR]

We show how the notion of fantastacks can be used to effectively desingularize binomial varieties defined over algebraically closed fields. In contrast to a desingularization via blow-ups in smooth centers, we drastically reduce the number of steps and the number of charts appearing along the process. Furthermore, we discuss how our considerations extend to a partial simultaneous normal crossings desingularization of finitely many binomial hypersurfaces. [ABSTRACT FROM AUTHOR]

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new invariant to isolated complete intersection singularities. We use this new invariant to obtain two characterization theorems for contact simple complete intersection singularities. [ABSTRACT FROM AUTHOR]

For i : Z ↪ X a closed immersion of smooth varieties, we study how the V-filtration along Z and the Hodge filtration on a mixed Hodge module M on X interact with each other. We also give a formula for the functors i ∗ , i ! in terms of this V-filtration. As applications, we obtain results on the Hodge filtration of monodromic mixed Hodge modules and we give a Hodge theoretic proof of Skoda's theorem on multiplier ideals. Finally, we use the results to study the Fourier–Laplace transform of a monodromic mixed Hodge module. [ABSTRACT FROM AUTHOR]

In this paper we present a constructive method to characterize ideals of the local ring O C n , 0 of germs of holomorphic functions at 0 ∈ C n which arise as the moduli ideal ⟨ f , m j (f) ⟩ , for some f ∈ m ⊂ O C n , 0 . A consequence of our characterization is an effective solution to a problem dating back to the 1980s, called the Reconstruction Problem of the hypersurface singularity from its moduli algebra. Our results work regardless of whether the hypersurface singularity is isolated or not. [ABSTRACT FROM AUTHOR]

We show a vanishing result for the first direct image of a proper birational morphism of normal quasi-excellent schemes with regular codomain. As a consequence, we deduce the birational invariance of H 1 (X , O X) for regular quasi-excellent schemes X. More generally, this is shown for quasi-excellent X which are S 3 and pseudo-rational in codimension 2. [ABSTRACT FROM AUTHOR]

We prove the abundance theorem for arithmetic klt threefold pairs whose closed point have residue characteristic greater than 5. As a consequence, we give a sufficient condition for the asymptotic invariance of plurigenera for certain families of singular surface pairs to hold in mixed characteristic. [ABSTRACT FROM AUTHOR]

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens' non-standard tight closure (Schoutens in Manuscr Math 111:379–412, 2003), ultra-tight closure, on ideals of a local domain R essentially of finite type over C . We prove that the ultra-test ideal τ u (R , a t) , the annihilator ideal of all ultra-tight closure relations of R, coincides with the multiplier ideal J (Spec R , a t) if R is normal Q -Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the weighted log canonical thresholds of holomorphic functions. We prove the ascending chain condition for certain weighted thresholds. [ABSTRACT FROM AUTHOR]

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in C n which is Lipschitz regular at infinity must be an affine linear subspace of C n . In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if X and Y are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then X is a complex algebraic set if and only if Y is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here. [ABSTRACT FROM AUTHOR]

For each n ≥ 3 we give examples of infinitesimally rigid projective manifolds of general type of dimension n with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover. [ABSTRACT FROM AUTHOR]

Given a resolution of rational singularities $\pi \colon {\tilde {X}} \to X$ over a field of characteristic zero, we use a Hodge-theoretic argument to prove that the image of the functor ${\mathbf {R}}\pi _*\colon {\mathbf {D}}^{\mathrm {b}}({\tilde {X}}) \to {\mathbf {D}}^{\mathrm {b}}(X)$ between bounded derived categories of coherent sheaves generates ${\mathbf {D}}^{\mathrm {b}}(X)$ as a triangulated category. This gives a weak version of the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21]. The same result is established more generally for proper (not necessarily birational) morphisms $\pi \colon {\tilde {X}} \to X$ , with ${\tilde {X}}$ smooth, satisfying ${\mathbf {R}}\pi _*({\mathcal {O}}_{\tilde {X}}) = {\mathcal {O}}_X$ .

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann. 271 (1985), 439–449] and S. Ishii [Math. Ann. 275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].

Let (V , 0) ⊂ (C n , 0) be a weighted homogeneous isolated hypersurface singularity. In this paper, we give explicit formulas of its k-th Milnor numbers and the k-th Tjurina numbers in terms of its weight type. Moreover, we propose a sharp lower bound conjecture for the k-th Tjurina numbers and verify this conjecture for binomial singularities. We also give a new characterization for the simple hypersurface singularities. [ABSTRACT FROM AUTHOR]

This paper proposes the use of F-split and globally F-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties. [ABSTRACT FROM AUTHOR]

The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I (a) ⊆ I b . Over a regular ring, a result by Ein–Lazarsfeld–Smith, Hochster–Huneke, and Ma–Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals—although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo–Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster–Huneke and Takagi. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if $(X,0)\subset (\mathbb {C}^{n},0)$ , $(Y,0)\subset (\mathbb {C}^{m},0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: if $X\subset \mathbb {C}^{n}$ , $Y\subset \mathbb {C}^{m}$ are algebraic sets and $\phi \colon X\to Y$ is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb {P}^{n+m}(\mathbb {C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$. [ABSTRACT FROM AUTHOR]

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$ -constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$ , for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W({\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W({\mathbb {C}}_{o})}_{m}$ is the relative space of $m$ -jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra. [ABSTRACT FROM AUTHOR]