Resolution à la Kronheimer of $$\mathbb {C}^3/\Gamma $$ singularities and the Monge–Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds $$Y^\Gamma $$ Y Γ that are supposed to be the crepant resolution of quotient singularities $$\mathbb {C}^3/\Gamma $$ C 3 / Γ with $$\Gamma $$ Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms $$\omega ^{2,1}$$ ω 2 , 1 . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on $$Y^\Gamma $$ Y Γ . We study the issue of Ricci-flat Kähler metrics on such resolutions $$Y^\Gamma $$ Y Γ , with particular attention to the case $$\Gamma =\mathbb {Z}_4$$ Γ = Z 4 . We advance the conjecture that on the exceptional divisor of $$Y^\Gamma $$ Y Γ the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on $$\mathrm{tot} K_{{\mathbb {W}P}[112]}$$ tot K W P [ 112 ] that we construct, i.e., the total space of the canonical bundle of the weighted projective space $${\mathbb {W}P}[112]$$ W P [ 112 ] , which is a partial resolution of $$\mathbb {C}^3/\mathbb {Z}_4$$ C 3 / Z 4 . For the full resolution, we have $$Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}$$ Y Z 4 = tot K F 2 , where $$\mathbb {F}_2$$ F 2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $$\mathbb {F}_2$$ F 2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.

Hilbert’s 16th problem on a period annulus and Nash space of arcs

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then $$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$ We prove this when $Z$ has isolated singularities and when it is a toric variety. We deduce that for a divisor $D$ with isolated rational singularities on a smooth complex $n$-dimensional variety $X$, the generation level of Saito’s Hodge filtration on the localization $\mathscr{O}_{X}(\ast D)$ is at most $n-3$.

Projections of del Pezzo surfaces and Calabi–Yau threefolds

We study the syzygetic structure of projections of del Pezzo surfaces in order to construct singular Calabi-Yau threefolds. By smoothing those threefolds,we obtain new examples of Calabi-Yau threefolds with Picard group of rank 1. We also give an example of type II primitive contraction whose exceptional divisor is the blow-up of the projective plane at a point.

Exceptional divisors that are not uniruled belong to the image of the Nash map

We prove that, ifXis a variety over an uncountable algebraically closed fieldkof characteristic zero, then any irreducible exceptional divisorEon a resolution of singularities ofXwhich is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs$X_\infty^{\mathrm{Sing}}$onXcentred in SingX. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.

ON CAMACHO–SAD'S THEOREM ABOUT THE EXISTENCE OF A SEPARATRIX

It is proved in Ann. Math. (2)115 (1982) 579–595 that, for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix (invariant analytic curve containing the origin). In Proc. Amer. Math. Soc.125 (1997) 2649–2650 a simple criterion was given to find, at each level of the blow-up, a singular point which leads to an analytical invariant curve. In this paper we prove shortly and strictly combinatorially, the existence of a separatrix, and show that for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix issuing from each connected component of the exceptional divisor of its nice blow-up with nodal corner points deleted.

A Remark on Partial Resolutions of 3-Dimensional Terminal Singularities

Let X be a 3-dimensional terminal singularity of index ≥ 2. We study projective birational morphisms ϕ: Y → X such that the exceptional divisor of ϕ consists of all prime divisors with discrepancies < 1 (resp. ≤ 1) over X.

Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

Let Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

Differential Forms and Resolutions on Certain Analytic Spaces II. Flat Resolutions

This paper gives another construction of (0, p)-forms on a complex analytic space and of the operator. This construction is independent of the one in [1] and apart from the general result of Section 1 of [1], it can be read independently. As in [1], the hypotheses on S are the following: S has normal singularities, its singular locus X is smooth, the exceptional divisor in a desingularization of S is irreducible.

On the topology of full non-degenerate complete intersection variety

Let h1(u),…, hk(u) be Laurent polynomials of m-variables and letbe a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).