The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

Photoswitchable Chiral Organocatalysts: Photocontrol of Enantioselective Reactions

This study presents recent advances in photoswitchable chiral organocatalysts and their applications in the photomodulation of enantioselective reactions. Under irradiation with an appropriate wavelength of light, the E/Z-photoisomerization of the photoresponsive units on the catalysts leads to the control of the catalytic activity and/or selectivity of the enantioselective reactions. Additionally, this study elucidates the design, synthesis, and catalytic application of the fabricated azobenzene BINOL-based photoswitchable chiral phase-transfer catalysts. This account will provide insights into the appropriate design of a photoswitchable chiral organocatalyst that can achieve both good enantioselectivity and photocontrol.