Roberto Svaldi
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Characterization of products of projective spaces via nef complexity Open
We define the nef complexity of a projective variety $X$. This invariant compares $\dim X+ρ(X)$ with the sum of the coefficients of nef partitions of $-K_X$. We prove that the nef complexity is non-negative and it is zero precisely for pro…
Variation of algebraically integrable adjoint foliated structures Open
Given a canonical algebraically integrable foliation on a klt projective variety, we study the variation of the ample models of the associated adjoint foliated structures with respect to the parameter. When the foliation is of general type…
Boundedness of some fibered K-trivial varieties Open
We prove that irreducible Calabi-Yau varieties of a fixed dimension, admitting a fibration by abelian varieties or primitive symplectic varieties of a fixed analytic deformation class, are birationally bounded. We prove that there are only…
On finite generation and boundedness of adjoint foliated structures Open
We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and para…
Boundedness of elliptic Calabi–Yau varieties with a rational section Open
We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical …
Minimal model program for algebraically integrable adjoint foliated structures Open
For $\mathbb Q$-factorial klt algebraically integrable adjoint foliated structures, we prove the cone theorem, the contraction theorem, and the existence of flips. Therefore, we deduce the existence of the minimal model program for such st…
Boundedness of elliptic Calabi–Yau threefolds Open
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 …
On the connectedness principle and dual complexes for generalized pairs Open
Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-({K_{X}} + B)$ nef over S . A conjecture, known as the Shokurov–Kollár connectedness principle, predicts that $f^{-1} (s) \cap \operatorname {\mathrm {Nklt}}…
Effective generation for foliated surfaces: Results and applications Open
We explore the birational structure and invariants of a foliated surface ( X , F ) (X,\mathcal{F}) in terms of the adjoint divisor K F + ϵ K X K_{\mathcal{F}}+\epsilon K_{X} , 0 < ϵ ≪ 1 0<\epsilon\ll 1 . We then establish a bound on the …
View article: The Jordan property for local fundamental groups
The Jordan property for local fundamental groups Open
We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index 1 covers for n-dimensional klt singularities. We give an application t…
Rational curves and strictly nef divisors on Calabi-Yau threefolds Open
We give a criterion for a nef divisor D to be semi-ample on a Calabi-Yau threefold X when D^3=0=c_2(X)\cdot D and c_3(X)\neq 0 . As a direct consequence, we show that on such a variety X , if D is strictly nef and \nu(D)\neq 1 , then D is …
Local and global applications of the Minimal Model Program for co-rank 1 foliations on threefolds Open
We provide several applications of the minimal model program to the local and global study of co-rank 1 foliations on threefolds. Locally, we prove a singular variant of Malgrange’s theorem, a classification of terminal foliation singulari…
Boundedness of elliptic Calabi-Yau threefolds Open
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 …
Rational Curves and Strictly nef Divisors on Calabi-Yau Threefolds Open
We give a criterion for a nef divisor $D$ to be semi-ample on a Calabi-Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$,…
A geometric characterization of toric singularities Open
Given a projective contraction $π\colon X\rightarrow Z$ and a log canonical pair $(X, B)$ such that $-(K_X+B)$ is nef over a neighborhood of a closed point $z\in Z$, one can define an invariant, the complexity of $(X, B)$ over $z \in Z$, c…
Effective generation for foliated surfaces: results and applications Open
We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+εK_X$, $0< ε\ll 1$. We then establish a bound on the automorphism group of an adjoint general type f…
Rational curves and strictly nef divisors on Calabi--Yau threefolds Open
We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ν(D)\neq 1$, t…
Boundedness of elliptic Calabi-Yau varieties with a rational section Open
We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical …
On the connectedness principle and dual complexes for generalized pairs Open
Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$. A conjecture, known as the Shokurov-Kollár connectedness principle, predicts that $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most t…
Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds Open
We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singula…
Effective algebraic integration in bounded genus Open
We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results in order to investigate the effective…
A geometric characterization of toric varieties Open
We prove a conjecture of Shokurov which characterizes toric varieties using log pairs.
On the birational boundedness of the bases of elliptically fibered CY's in low dimension Open
I will discuss joint work with Gabriele Di Cerbo on boundedness of Calabi-Yau pairs. Given an elliptically fibered Calabi-Yau manifold, the base of the fibration naturally carries the structure of a Calabi-Yau pair, that is, there exists a…
Hyperbolicity for log pairs Open
A classical result in birational geometry, Mori’s Cone Theorem, implies that if the canonical bundle of a variety X is not nef then X contains rational curves. This is the starting point of the so-called Minimal Model Program. In particula…