Kodaira Dimension Research Articles

Autoequivalences of blow‐ups of minimal surfaces

AbstractLet be the blow‐up of in a finite set of very general points. We deduce from the work of Uehara [Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529–3547] that has only standard autoequivalences, no non‐trivial Fourier–Mukai partners, and admits no spherical objects. If is the blow‐up of in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if is a blow‐up of finitely many points in a minimal surface of non‐negative Kodaira dimension which contains no ‐curves. Independently, we characterize spherical objects on blow‐ups of minimal surfaces of positive Kodaira dimension.

The association between toenail metals and extracellular MicroRNAs (ex-miRNAs) among the participants of the Normative Aging study (NAS)

BackgroundMechanistic studies of the effects of environmental risk factors have been exploring the potential role of microRNA(miRNAs) as a possible pathway to clinical disease. In this study we examine whether levels of toenail metals are associated with changes in extracellular miRNA(ex-miRNA) expression. MethodsWe used data derived from the Normative Aging Study from 1996 to 2014 to conduct our analyses. We looked at associations between measured toenail metals: arsenic, cadmium, lead, manganese, and mercury and 282 ex-miRNAs in this population using canonical correlation analyses (CCAs) and longitudinal median regression. We adjusted for covariates such as age, education, body mass index, drinking and smoking behaviors, diabetes, and where available, seafood consumption. The p-values obtained from regression analyses were corrected for multiple comparisons. Ex-miRNAs identified to be associated with toenail metal levels were further examined using pathway analyses. ResultsOur dataset included 937 observations from 589 men with an average age of 72.9 years at baseline. Both our correlation and regression analyses identified lead and cadmium as exposures most strongly associated with ex-miRNA expression. Numerous ex-miRNAs were identified as being associated with toenail metal levels. miR-27b-3p, in particular, was found to have high correlation with the first canonical dimension in the CCA and was significantly associated with cadmium in the regression analysis. Pathway analyses revealed messenger RNA (mRNA) targets for the ex-miRNAs that were associated with a number of clinical disorders including cancer, cardiovascular disease, and neurological disorders, etc. ConclusionToenail metals were associated with changes in ex-miRNA levels in both correlational and regression analyses. The ex-miRNAs identified can be linked to a variety of clinical disorders. Further studies are required to validate these findings.

Left-invariant almost complex structures on the higher dimensional Kodaira–Thurston manifolds

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds.

Boundedness of elliptic Calabi–Yau threefolds

We show that elliptic Calabi–Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2 , upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these hypotheses are necessary to prove the boundedness of such a family.

Mukai models and Borcherds products

abstract: Let ${\Fgn}$ be the moduli space of $n$-pointed $K3$ surfaces of genus $g$ with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on ${\Fgn}$ and the ring of certain orthogonal modular forms, and give applications to the birational type of ${\Fgn}$. We prove that the Kodaira dimension of ${\Fgn}$ stabilizes to $19$ when $n$ is sufficiently large. Then we use explicit Borcherds products to find a lower bound of $n$ where ${\Fgn}$ has nonnegative Kodaira dimension, and compare this with an upper bound where ${\Fgn}$ is unirational or uniruled using Mukai models of $K3$ surfaces in $g\leq 20$. This reveals the exact transition point of Kodaira dimension in some~$g$.

On varieties whose general surface section has negative Kodaira dimension

On varieties whose general surface section has negative Kodaira dimension

The nonabelian Brill–Noether divisor on ℳ13 and the Kodaira dimension of ℛ13

The nonabelian Brill–Noether divisor on ℳ13 and the Kodaira dimension of ℛ13

Reflective obstructions of unitary modular varieties

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than 138. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula.

Kawaguchi–Silverman conjecture on automorphisms of projective threefolds

Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang, and Hu and Li, we may reduce Kawaguchi–Silverman conjecture for automorphisms [Formula: see text] on normal projective threefolds [Formula: see text] with either the canonical divisor [Formula: see text] is trivial or negative Kodaira dimension to the following two cases: (i) [Formula: see text] is a primitively automorphism of a weak Calabi–Yau threefold, (ii) [Formula: see text] is a rationally connected threefold. And we prove Kawaguchi–Silverman conjecture is true for automorphisms of normal projective varieties [Formula: see text] with the irregularity [Formula: see text]. Finally, we discuss Kawaguchi–Silverman conjecture on normal projective varieties with Picard number two.

On the Kodaira dimension of moduli spaces of Abelian differentials

On the Kodaira dimension of moduli spaces of Abelian differentials

Kodaira dimensions of almost complex manifolds II

Kodaira dimensions of almost complex manifolds II

Dark sector effective field theory

We introduce the effective field theory of two different light dark particles interacting with the standard model (SM) light states in a single vertex, termed dark sector effective field theory (DSEFT). We focus on the new light particles with spin up to 1 and being real in essence, namely, new real scalars ϕ and S, Majorana fermions χ and ψ, and real vectors Xμ and Vμ. In the framework of low energy effective field theory with QED and QCD symmetry, the DSEFT can be classified into six categories, including the scalar-scalar-SM (ϕS-SM), fermion-fermion-SM (χψ-SM), vector-vector-SM (XV-SM), scalar-fermion-SM (ϕχ-SM), scalar-vector-SM (ϕX-SM), and fermion-vector-SM (χX-SM) cases. For each case, we construct the effective operator basis up to canonical dimension 7, which will cover most interesting phenomenology at low energy. As a phenomenological example, we investigate the longstanding neutron lifetime anomaly through the neutron dark decay modes n → χϕ or χX from the effective interactions in the fermion-scalar-SM or fermion-vector-SM case. When treating the light fermion as a dark matter candidate, we also explore the constraints from DM-neutron annihilation signal at Super-Kamiokande. We find the neutron dark decay in each scenario can accommodate the anomaly, at the same time, without contradicting with the Super-Kamiokande limit.

Birational geometry of smooth families of varieties admitting good minimal models

We study families of projective manifolds with good minimal models. After constructing a suitable moduli functor for polarized varieties with canonical singularities, we show that, if not birationally isotrivial, the base spaces of such families support subsheaves of log-pluridifferentials with positive Kodaira dimension. Consequently we prove that, over special base schemes, families of this type can only be birationally isotrivial and, as a result, confirm a conjecture of Kebekus and Kovács.

The birational invariants of Lins Neto’s foliations

Lins Neto [Ann. Sci. École Norm. Sup. (4) 35 (2002), pp. 231–266] constructed families of foliations which are counterexamples to Poincaré’s Problem and Painlevé’s Problem. We will determine the minimal models of these families of foliations, calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. We prove that the slopes of Lins Neto’s foliations are at least 6, and their limits are bigger than 7 7 .

Kodaira Dimension of Fibrations Over Abelian Varieties

Abstract We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs.

On the behavior of the Kodaira dimension under smooth morphisms

On the behavior of the Kodaira dimension under smooth morphisms

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

Almost complex parallelizable manifolds: Kodaira dimension and special structures

We study the Kodaira dimension of a real parallelizable manifold M, with an almost complex structure J in standard form with respect to a given parallelism. For X=(M,J)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = (M, J)$$\\end{document} we give conditions under which kod(X)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{kod}\\,}}(X) = 0$$\\end{document}. We provide examples in the case M=G×G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G \ imes G$$\\end{document}, where G is a compact connected real Lie group. Finally we describe geometrical properties of real parallelizable manifolds in the framework of statistical geometry.

Bounded cohomology property on a smooth projective surface with Picard number two

We say a smooth projective surface X satisfies the bounded cohomology property if there exists a positive constant such that for every prime divisor C on X. Let the closed Mori cone such that and with are some curves on X. If either (i) the Kodaira dimension or (ii) , the irregularity and the Iitaka dimension , then we prove that X satisfies the bounded cohomology property.

Addendum: Weyl cocycles, (1986 Class. Quantum Grav. 3 635)

Weyl 0- and 1-cocycles of canonical dimension 6 in six dimensions, which were computed earlier in Bonora et al (1986 Class. Quantum Grav. 3 635), are recalculated from scratch. The analysis yields five Weyl invariants (0-cocycles), instead of four, and the same four non-trivial 1-cocycles (possible trace anomalies), like in that reference (up to the correction of one typo).