Related papers: Log canonical $3$-fold complements

We show that flat families of stable 3-folds do not lead to proper moduli spaces in any characteristic $p>0$. As a byproduct, we obtain log canonical 4-fold pairs, whose log canonical centers are not weakly normal.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence,…

Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational…

We construct Calabi-Yau 3-folds as orbifolds embedded in weighted projective space in codimension 4. For each Hilbert series that is realised, there are at least two different components of Calabi-Yau 3-folds.

In this article we show that if $(X, \Delta)$ is a log canonical compact K\"ahler threefold pair such that $K_X+\Delta$ is nef and the numerical dimension $\nu(X, K_X+\Delta)=2$, then $K_X+\Delta$ is semi-ample. This result combined with…

Let $f: X \to Z$ be a fibration from a normal projective variety $X$ of dimension $n$ onto a normal curve $Z$ over a perfect field of characteristic $p>2$. Let $(X, B)$ be a dlt pair such that the induced pair on a general fibre is log…

We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w…

In this paper, we prove the cone theorem and the contraction theorem for pairs $(X, B)$, where $X$ is a normal variety and $B$ is an effective $\mathbb R$-divisor on $X$ such that $K_X+B$ is $\mathbb R$-Cartier.

Let $(X,\Delta)$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,\Delta)$ and suppose that there exists a torus $\mathbb{T}\subseteq\operatorname{Aut}(X)$…

For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic…

Recent results on duality between string theories and connectedness of their moduli spaces seem to go a long way toward establishing the uniqueness of an underlying theory. For the large class of Calabi-Yau 3-folds that can be embedded as…

In this article we show that the Log Minimal Model Program holds for $\mathbb{Q}$-factorial lc pair $(X,\Delta)$ with $X$ being a compact K\"ahler $3$-fold having only klt singularities.

In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log…

A Calabi-Yau threefold is called of type K if it admits an \'etale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper, based on Oguiso-Sakurai's fundamental work, we provide the full classification…

This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce $K$-theoretic $\mathrm{DT}, \mathrm{PT}_0, \mathrm{PT}_1$ invariants and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$…

Generalizing the work of Sen, we analyze special points in the moduli space of the compactification of the F-theory on elliptically fibered Calabi-Yau threefolds where the coupling remains constant. These contain points where they can be…

Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$…

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds…

In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when $X/G$ has terminal singularities. We prove that, if $G$ is cyclic of prime order and $X/G$ has…

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau…