On minimal log discrepancies and kollár components

In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

5d and 4d SCFTs: canonical singularities, trinions and S-dualities

Canonical threefold singularities in M-theory and Type IIB string theory give rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. In this paper, we study canonical hypersurface singularities whose resolutions contain residual terminal singularities and/or 3-cycles. We focus on a certain class of ‘trinion’ singularities which exhibit these properties. In Type IIB, they give rise to 4d $$ \mathcal{N} $$ N = 2 SCFTs that we call $$ {D}_p^b $$ D p b (G)-trinions, which are marginal gaugings of three SCFTs with G flavor symmetry. In order to understand the 5d physics of these trinion singularities in M-theory, we reduce these 4d and 5d SCFTs to 3d $$ \mathcal{N} $$ N = 4 theories, thus determining the electric and magnetic quivers (or, more generally, quiverines). In M-theory, residual terminal singularities give rise to free sectors of massless hypermultiplets, which often are discretely gauged. These free sectors appear as ‘ugly’ components of the magnetic quiver of the 5d SCFT. The 3-cycles in the crepant resolution also give rise to free hypermultiplets, but their physics is more subtle, and their presence renders the magnetic quiver ‘bad’. We propose a way to redeem the badness of these quivers using a class $$ \mathcal{S} $$ S realization. We also discover new S-dualities between different $$ {D}_p^b $$ D p b (G)-trinions. For instance, a certain E8 gauging of the E8 Minahan-Nemeschansky theory is S-dual to an E8-shaped Lagrangian quiver SCFT.

The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Let [Formula: see text] be an elliptic surface over [Formula: see text] with [Formula: see text], and [Formula: see text] be the moduli scheme of rank-two stable sheaves [Formula: see text] on [Formula: see text] with [Formula: see text] in [Formula: see text]. We look into defining equations of [Formula: see text] at its singularity [Formula: see text], partly because if [Formula: see text] admits only canonical singularities, then the Kodaira dimension [Formula: see text] can be calculated. We show the following: (A) [Formula: see text] is at worst canonical singularity of [Formula: see text] if the restriction of [Formula: see text] to the generic fiber of [Formula: see text] has no rank-one subsheaf, and if the number of multiple fibers of [Formula: see text] is a few. (B) We obtain that [Formula: see text] and the Iitaka program of [Formula: see text] can be described in purely moduli-theoretic way for [Formula: see text], when [Formula: see text], [Formula: see text] has just two multiple fibers, and one of its multiplicities equals [Formula: see text]. (C) On the other hand, when [Formula: see text] has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether [Formula: see text] is at worst canonical singularity or not.

On birational boundedness of foliated surfaces

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}}, then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}}, then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N}.On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.

On the Nonnegativity of Stringy Hodge Numbers

We study the nonnegativity of stringy Hodge numbers of a projective variety with Gorenstein canonical singularities, which was conjectured by Batyrev. We prove that the $(p,1)$-stringy Hodge numbers are nonnegative, and for three-folds we obtain new results about the stringy Hodge diamond, which hold even when the stringy $E$-function is not a polynomial. We also use the Decomposition Theorem and mixed Hodge theory to prove Batyrev’s conjecture for a class of four-folds.