Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems

In the 1st part of the paper, we study a Fujita-type conjecture by Popa and Schnell and obtain an effective bound on the generic global generation of direct images of twisted pluricanonical bundles. We also point out the relation between the Seshadri constant and the optimal bound. In the 2nd part, we give an affirmative answer to a question by Demailly, Peternell, and Schneider in a more general setting. As byproducts of our proofs, we extend a result by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal settings and refine a theorem by Broustet and Pacienza on the rational connectedness of the image.

On the existence of certain weak Fano threefolds of Picard number two

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number $1$ with the exception of $12$ numerical cases.

Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to speciûc toric subvarieties and expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as corresponding Laurent polynomials.We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so–called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi–Yau varieties.