arXiv subjects
Publications and source records attributed to Stefano Filipazzi.
We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T 0$, where $n$ is an uniform natural number. This means that every $3$-fold of log Calabi-Yau type can be turned into a log Calabi-Yau pair in an effective way.
We prove boundedness of global strong $(δ,n)$-complements for generalized $ε$-log canonical pairs of Fano type. We also prove some partial results towards boundedness of local strong $(δ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(δ,n)$-complements implies a conjecture due to M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fiber space.
In this note, using methods introduced by Hacon, McKernan and Xu, we study the accumulation points of volumes of varieties of log general type. First, we show that, if the set of boundary coefficients $Λ$ is DCC, closed under limits and contains 1, then also the corresponding set of volumes is DCC and closed under limits. Then, we consider the case of $ε$-log canonical varieties, for $0 < ε< 1$. In this situation, we prove that, if $Λ$ is finite, then the corresponding set of volumes is discrete.
In this note, we extend the theories of the canonical bundle formula and adjunction to the case of generalized pairs. As an application, we study a particular case of a conjecture by Prokhorov and Shokurov.
We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum arXiv:alg-geom/9301001, and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the Berglund-Hübsch mirror symmetry setup for this example arXiv:hep-th/9201014. We then follow the approach of Krawitz to propose an explicit mirror map arXiv:0906.0796.
In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results due to Alexeev [Ale94].
We show that there exist smooth surfaces violating Generic Vanishing in any characteristic $p \geq 3$. As a corollary, we recover a result of Hacon and Kovács, producing counterexamples to Generic Vanishing in dimension 3 and higher.