Related papers: Weak Landau-Ginzburg models for smooth Fano threef…
We refine the classification of weak Fano threefolds with sextic del Pezzo fibrations by considering the Hodge numbers of them. By the refined classification result, such threefolds are classified into 17 cases. The main result of this…
We show how Calabi-Yau hypersurface families arising from Batyrev's construction can be resolved and compactified using a type of fan more general than an MPCP resolution. This can lead to smooth projective compactifications that are not…
We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a semi-norm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f.…
By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
We show the validity of two special cases of the four-dimensional Minimal Model Program in characteristic $p>5$: for contractions to $\mathbb{Q}$-factorial fourfolds and in families over curves ("semi-stable mmp"). We also provide their…
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt…
We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect…
We describe mirror symmetry for weak toric Fano manifolds as an equivalence of D-modules equipped with certain filtrations. We discuss in particular the logarithmic degeneration behavior at the large radius limit point, and express the…
There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For…
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds;…
Ginzburg-Landau model with two order parameters appears in many condensed-matter problems. However, even for scalar order parameters, the most general U(1)-symmetric Landau potential with all quadratic and quartic terms contains 13…
We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarised varieties. They are given by moduli spaces of…
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has…
We construct 16 new examples of Calabi--Yau threefolds with Picard group of rank 1. Each of these examples is obtained by smoothing the image of a primitive contraction with exceptional divisor being a del Pezzo surface of degree 6, 7 or…
Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional…
In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at…
The results of A.Chiodo, Y.Ruan and M.Krawitz associate the mirror partner Calabi-Yau variety $X$ to a Landau--Ginzburg orbifold $(f,G)$ if $f$ is an invertible polynomial satisfying Calabi-Yau condition and the group $G$ is a diagonal…
In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral…
We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to…
We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds…