Singular spaces with trivial canonical class

Pathologies of the Hilbert scheme of points of a supersingular Enriques surface

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.

We prove that a geometrically integral smooth projective 3‐fold with nef anti‐canonical class and negative Kodaira dimension over a finite field of characteristic and cardinality has a rational point. Additionally, under the same assumptions on and , we show that a smooth projective 3‐fold with trivial canonical class and non‐zero first Betti number has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi–Yau 3‐fold pairs over perfect fields.

Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X)$ of a compact Kahler manifold with $K_X$ nef is either trivial or a complex torus.

In this paper, a new holomorphic invariant is defined on a compact Kähler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the Futaki invariant. We prove that this invariant is an obstruction t

We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.

These varieties are conjectured to be abelian varieties up to finite etale coverings. This conjecture is derived from an affirmative answer to the abundance conjecture in minimal model theory. In particular, this is true for n=3.

In the first section, we give a quick introduction to the birational classification theory due to Enriques-Kodaira-Shafarevich-Iitaka, which divides the n-dimensional algebraic varieties into n + 2 disjoint classes according to a single invariant called the “Kodaira dimension” k. A nice thing of the theory is that a variety of intermediate Kodaira dimension has a canonical structure of a fibre space unique up to birational equivalence (“Iitaka fibration”).This classification would be incorporated to the Minimal Model Program through the “abundance conjecture”, which asserts that the canonical divisor of a minimal variety should be semiample, meaning in particular that a numerical invariant v(X) computes the Kodaira dimension k(X). Some easy cases where the abundance conjecture is actually verified are discussed in Section 2. Along with the minimal model theory, the existence of Iitaka fibration is also essential in the argument.In Section 3, the non-negativity of the Kodaira dimension of a minimal threefold is proved. The key to the proof is the pseudo-effectivity of c2 proved in Lecture III. We are exceptionally lucky in this case, because the Todd classes involve only c1 and c2 in dimension three.By the result in Section 3, we can find an effective divisor S in a pluricanonical linear system on a minimal threefold. The proof of the abundance conjecture relies on the analysis of linear systems on S. Principal difficulties are caused from the fact that S may be highly reducible and non-reduced. The complete proof of the three-dimensional abundance conjecture requires fairly technical materials (log-minimal models etc.), which would be beyond the scope of these lectures intended for non-specialists. Instead, Section 4 gives a proof in easier cases where S is smooth. Still the discussion involves the heart of the idea and, hopefully, will illustrate guiding principles toward the proof in general cases.KeywordsRational CurfEffective DivisorSmooth Projective VarietyCartier DivisorFibre SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Framed temporal logic programming

There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space (mathcal {R}_0, p), where mathcal {R}_0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parameters (mathcal {R}_0, p) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters.

We introduce an appropriate class of higher order linear ordinary differential operators with a large parameter and with simple poles in their coefficients, and study their structure from the viewpoint of exact WKB analysis, i.e., the WKB analysis based on the Borel resummation. Our main result is that an operator in the class is expressed as a product of two operators Q and R, with Q being irrelevant to the Stokes geometry and with R being a second order simple-pole type operator studied by Koike ([Ko1], [Ko2], [Ko3]). This decomposition theorem gives us a connection formula for WKB solutions near the simple pole in question, and it is used to explain the background mechanism of some intriguing phenomenon we encounter in the study of a particular third order operator. Some discussions on the scope of the future development of the theory are included.

The goal of this work is to study and prove some of the main results of Obstruction Theory, as well as to present some possible applications. The proof of these results depends on the development of several prerequisites along the way, like the notions of free and pointed homotopy, H-groups and H-cogroups, homotopy groups and locally trivial bundles. This development culminates in the proof that the problem of extending maps and sections over the skeletons of a CW-complex is controlled by a cohomological invariant. This result is then used to construct the characteristic classes associated with a vector bundle, and also to define the local Euler obstruction of a point in a singular space.

A unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.