Resolution Of Singularities Research Articles

Bouncing evolution in a model of loop quantum gravity

To understand the dynamics of loop quantum gravity, the deparametrized model of gravity coupled to a scalar field is studied in a simple case, where the graph underlying the spin network basis is one loop based at a single vertex. The Hamiltonian operator $\hat{H}_v$ is chosen to be graph-preserving, and the matrix elements of $\hat{H}_v$ are explicitly worked out in a suitable basis. The non-trivial Euclidean part $\hat{H}_v^E$ of $\hat{H}_v$ is studied in details. It turns out that by choosing a specific symmetrization of $\hat{H}_v^E$, the dynamics driven by the Hamiltonian gives a picture of bouncing evolution. Our result in the model of full loop quantum gravity gives a significant echo of the well-known quantum bounce in the symmetry-reduced model of loop quantum cosmology, which indicates a closed relation between singularity resolution and quantum geometry.

The bulk-to-boundary propagator in black hole microstate backgrounds

First-quantized propagation in quantum gravitational AdS3 backgrounds can be exactly reconstructed using CFT2 data and Virasoro symmetry. We develop methods to compute the bulk-to-boundary propagator in a black hole microstate, leftlangle {phi}_L{mathcal{O}}_L{mathcal{O}}_H{mathcal{O}}_Hrightrangle , at finite central charge. As a first application, we show that the semiclassical theory on the Euclidean BTZ solution sharply disagrees with the exact description, as expected based on the resolution of forbidden thermal singularities, though this effect may appear exponentially small for physical observers.

On a Resolution of Singularities with Two Strata

Let X be a complex, irreducible, quasi-projective variety, and $$\pi :{\widetilde{X}}\rightarrow X$$ a resolution of singularities of X. Assume that the singular locus $${\text {Sing}}(X)$$ of X is smooth, that the induced map $$\pi ^{-1}({\text {Sing}}(X))\rightarrow {\text {Sing}}(X)$$ is a smooth fibration admitting a cohomology extension of the fiber, and that $$\pi ^{-1}({\text {Sing}}(X))$$ has a negative normal bundle in $${\widetilde{X}}$$ . We present a very short and explicit proof of the Decomposition Theorem for $$\pi $$ , providing a way to compute the intersection cohomology of X by means of the cohomology of $${\widetilde{X}}$$ and of $$\pi ^{-1}({\text {Sing}}(X))$$ . Our result applies to special Schubert varieties with two strata, even if $$\pi $$ is non-small. And to certain hypersurfaces of $${\mathbb {P}}^5$$ with one-dimensional singular locus.

Phases of 5d SCFTs from M-/F-theory on non-flat fibrations

We initiate the systematic investigation of non-flat resolutions of non-minimal singularities in elliptically fibered Calabi-Yau threefolds. Compactification of M-theory on these geometries provides an alternative approach to studying phases of five-dimensional superconformal field theories (5d SCFTs). We argue that such resolutions capture non-trivial holonomies in the circle reduction of the 6d conformal matter theory that is the F-theory interpretation of the singular fibration. As these holonomies become mass deformations in the 5d theory, non-flat resolutions furnish a novel method in the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly pleasant aspect of this proposal is the explicit embedding of the 5d SCFT’s enhanced flavor group inside that of the parent 6d SCFT, which can be read off from the geometry. We demonstrate these features in toric examples which realize 5d theories up to rank four.

Generic absence of strong singularities and geodesic completeness in modified loop quantum cosmologies

Different regularizations of the Hamiltonian constraint in loop quantum cosmology (LQC) yield modified loop quantum cosmologies, namely mLQC-I and mLQC-II, which lead to qualitatively different Planck scale physics. We perform a comprehensive analysis of resolution of various singularities in these modified loop cosmologies using effective spacetime description and compare with earlier results in standard LQC. We show that the volume remains non-zero and finite in finite time evolution for all considered loop cosmological models. Interestingly, even though expansion scalar and energy density are bounded due to quantum geometry, curvature invariants can still potentially diverge due to pressure singularities at a finite volume. These divergences are shown to be harmless since geodesic evolution does not break down and no strong singularities are present in the effective spacetimes of loop cosmologies. Using a phenomenological matter model, various types of exotic strong and weak singularities, including big rip, sudden, big freeze and type-IV singularities, are studied. We show that as in standard LQC, big rip and big freeze singularities are resolved in mLQC-I and mLQC-II, but quantum geometric effects do not resolve sudden and type-IV singularities.

Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

Let G G be a profinite group. A strongly admissible smooth representation ϱ \varrho of G G over C {\mathbb {C}} decomposes as a direct sum ϱ ≅ ⨁ π ∈ Irr ( G ) m π ( ϱ ) π \varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \, \pi of irreducible representations with finite multiplicities m π ( ϱ ) m_\pi (\varrho ) such that, for every positive integer n n , the number r n ( ϱ ) r_n(\varrho ) of irreducible constituents of dimension n n is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ ζ ϱ ( s ) = ∑ n = 1 ∞ r n ( ϱ ) n − s = ∑ π ∈ Irr ( G ) m π ( ϱ ) ( dim ⁡ π ) s \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation ϱ \varrho . Our primary focus is on representations ϱ = Ind H G ( σ ) \varrho = \text {Ind}_H^G(\sigma ) of compact p p -adic Lie groups G G that arise from finite-dimensional representations σ \sigma of closed subgroups H H via the induction functor. In addition to a series of foundational results—including a description in terms of p p -adic integrals—we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro- p p groups. A key ingredient of our proof is Hironaka’s resolution of singularities, which yields formulae of Denef type for the relevant zeta functions. In some detail we consider representations of open compact subgroups of reductive p p -adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees, and (ii) the p p -adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.

Witt differentials in the h-topology

For sheaves of differential forms of the de Rham–Witt complex for regular varieties over a perfect field of positive characteristic p we prove unconditional descent in cohomological dimension 0 with respect to Voevodsky's h-topology. Under resolution of singularities we obtain full cohomological descent. Our approach follows recent work of Huber–Jörder and Huber–Kebekus–Kelly on sheaves of differential forms.

Resolution of singularities of arithmetical threefolds

We prove Grothendieck's conjecture on Resolution of Singularities for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X=SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective surface X/F has a proper and flat model X over O which is everywhere regular.

Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles

We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety$X$and any resolution of singularities, any vector bundle on the resolution that appears to come from$X$numerically, does indeed come from $X$. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

On the rationality problem for quadric bundles

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective n-space are not stably rational if r lies in the interval from $2^{n-1}-1$ to $2^{n}-2$. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Th\'el\`ene--Pirutka which avoids universally $CH_0$-trivial resolutions of singularities.

Bouncing unitary cosmology I. Mini-superspace general solution

We offer a new proposal for cosmic singularity resolution based upon a quantum cosmology with a unitary bounce. This proposal is illustrated via a novel quantization of a mini-superspace model in which there can be superpositions of the cosmological constant. This possibility leads to a finite, bouncing unitary cosmology. Whereas the usual Wheeler–DeWitt cosmology generically displays pathological behaviour in terms of non-finite expectation values and non-unitary dynamics, the finiteness and unitarity of our model are formally guaranteed. For classically singular models with a massless scalar field and cosmological constant, we show that well-behaved quantum observables can be constructed and generic solutions to the universal Schrödinger equation are singularity-free. Generic solutions of our model displays novel features including: (i) superpositions of values of the cosmological constant; (ii) universal effective physics due to non-trivial self-adjoint extensions of the Hamiltonian; and (iii) bound ‘Efimov universe’ states for negative cosmological constant. The last feature provides a new platform for quantum simulation of the early universe. A companion paper provides detailed interpretation and analysis of particular cosmological solutions that display a cosmic bounce due to quantum gravitational effects, a well-defined FLRW limit far from the bounce, and a semi-classical turnaround point in the dynamics of the scalar field which resembles an effective inflationary epoch.

Bouncing unitary cosmology II. Mini-superspace phenomenology

A companion paper provides a proposal for cosmic singularity resolution based upon general features of a bouncing unitary cosmological model in the mini-superspace approximation. This paper analyses novel phenomenology that can be identified within particular solutions of that model. First, we justify our choice of particular solutions based upon a clearly articulated and observationally-motivated principle. Second, we demonstrate that the chosen solutions follow a classical mini-superspace cosmology before smoothly bouncing off the classically singular region. Third, and most significantly, we identify a ‘Rayleigh-scattering’ limit for physically reasonable choices of parameters within which the solutions display effective behaviour that is insensitive to the details of rapidly oscillating Planck-scale physics. This effective physics is found to be compatible with an effective period of cosmic inflation well below the Planck scale. The detailed effective physics of this Rayleigh-scattering limit is provided via: (i) an exact analytical treatment of the model in the de Sitter limit; and (ii) numerical solutions of the full model.

On Archimedean zeta functions and Newton polyhedra

Let $f$ be a polynomial function over the complex numbers and let $\phi$ be a smooth function over $\mathbb{C}$ with compact support. When $f$ is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to $f$ and $\phi$. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to $f$. More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, and necessary in the study of local zeta functions through resolution of singularities. Our results refine the corresponding results of Varchenko and generalize the results of Denef and Sargos in the real case, to the complex setting.

Domain Decomposition for Quasi-Periodic Scattering by Layered Media via Robust Boundary-Integral Equations at All Frequencies

We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.

Semi-steady non-commutative crepant resolutions via regular dimer models

A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we detect another nice property on NCCRs that characterizes square dimer models. We call such NCCRs semi-steady NCCRs, and study their properties.

Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals

Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L an , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L an and that the non-archimedean Monge–Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

A higher-dimensional generalization of Mumford's rational pullback for Weil divisors

Mumford defined a rational pullback for Weil divisors on normal surfaces, which is linear, respects effectivity, and satisfies the projection formula. In higher dimensions, the existence of small resolutions of singularities precludes such general results. We single out a higher-dimensional situation that resembles the surface case and show for it that a rational pullback for Weil divisors exists, which is also linear, respects effectivity, and satisfies the projection formula.

The slice spectral sequence for singular schemes and applications

We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.

Classification of double octic Calabi–Yau threefolds with h1,2 ≤ 1 defined by an arrangement of eight planes

In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

On quasi-purity of the branch locus

Let $k$ be a field, $K/k$ finitely generated and $L/K$ a finite, separable extension. We show that the existence of a $k$-valuation on $L$ which ramifies in $L/K$ implies the existence of a normal model $X$ of $K$ and a prime divisor $D$ on the normalization $X_L$ of $X$ in $L$ which ramifies in the scheme morphism $X_L \rightarrow X$. Assuming the existence of a regular, proper model $X$ of $K$, this is a straight-forward consequence of the Zariski-Nagata theorem on the purity of the branch locus. We avoid assumptions on resolution of singularities by using M. Temkin's inseparable local uniformization theorem.