Joaquín Moraga

Bio: Joaquín Moraga is an academic researcher from Princeton University. The author has contributed to research in topics: Mathematics & Gravitational singularity. The author has an hindex of 7, co-authored 38 publications receiving 166 citations. Previous affiliations of Joaquín Moraga include University of Utah.

On weak Zariski decompositions and termination of flips

Abstract: We prove that termination of lower dimensional flips for generalized klt pairs implies termination of flips for log canonical generalized pairs with a weak Zariski decomposition. Moreover, we prove that the existence of weak Zariski decompositions for pseudo-effective generalized klt pairs implies the existence of minimal models for such pairs.

On minimal log discrepancies and Koll\'ar components.

Abstract: In this article we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities, with standard coefficients, which admit an $\epsilon$-plt blow-up have minimal log discrepancies belonging to a finite set which only depends on $d,a$ and $\epsilon$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Koll\'ar components.

The Jordan property for local fundamental groups

Abstract: We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.

Termination of pseudo-effective 4-fold flips

Abstract: Let $(X,\Delta)$ be a complex log canonical $4$-fold, with $K_X+\Delta$ a pseudo-effective $\mathbb{Q}$-divisor. We prove that any sequence of $(K_X+\Delta)$-flips terminates.

Log canonical $3$-fold complements

Abstract: We expand the theory of log canonical $3$-fold complements. More precisely, fix a set $\Lambda \subset \mathbb{Q}$ satisfying the descending chain condition with $\overline{\Lambda} \subset \mathbb{Q}$, and let $(X,B+B')$ be a log canonical 3-fold with $\mathrm{coeff}(B) \in \Lambda$ and $K_X + B$ $\mathbb{Q}$-Cartier. Then, there exists a natural number $n$, only depending on $\Lambda$, such that the following holds. Given a contraction $f \colon X \rightarrow T$ and $t \in T$ with $K_X + B + B' \sim_\mathbb{Q} 0$ over $t$, there exists $\Gamma \geq 0$ such that $\Gamma \sim -n(K_X + B)$ over $t \in T$, and $(X,B+\Gamma/n)$ is log canonical.

ACC for minimal log discrepancies of exceptional singularities.

Abstract: We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,\Gamma_0)$-decomposable $\mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.

On minimal log discrepancies

Abstract: An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower semi-continuity behaviour for varieties of dimension at most 3 and for toric varieties of arbitrary dimension.