Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples

This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.     

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-foldswith conical singularities and their desingularizations. Inour first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181]we constructed the desingularization of the conically singularmanifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau3-fold Y into M₀ at the singular point, thus obtaining a 1-parameterfamily of compact, non-singular Calabi–Yau 3-folds M_tfor small t > 0. During the gluing process one may encountera kind of cohomological obstruction to defining a 3-form _t onM_t which interpolates between the 3-form ₀ on M₀ and the scaled3-form t³ _Y on Y if the rate at which the AC Calabi–Yau3-fold Y converges to the Calabi–Yau cone is equal to– 3. The first paper [3] studied the simpler case <–3 where there is no obstruction. This paper extends theresult in the first one by considering a more complicated situtationwhen = –3. Assuming the existence of singular Calabi–Yaumetrics on compact complex 3-folds with ordinary double points,our result in this paper can be applied to repairing such kindsof singularities, which is an analytic version of Friedman'sresult giving necessary and sufficient conditions for smoothingordinary double points.     

A Combined Experimental and Theoretical Study of the Versatile Reactivity of an Oxocerium(IV) Complex: Concerted Versus Reductive Addition

A combined experimental and theoretical investigation on the cerium(IV) oxo complex [(L_OEt)₂Ce(=O)(H₂O)] ⋅ MeC(O)NH₂ ( 1 ; L_OEt⁻=[Co(η⁵-C₅H₅){P(O)(OEt)₂}₃]⁻) demonstrates that the intermediate spin-state nature of the ground state of the cerium complex is responsible for the versatility of its reactivity towards small molecules such as CO, CO₂, SO₂, and NO. CASSCF calculations together with magnetic susceptibility measurements indicate that the ground state of the cerium complex is of multiconfigurational character and comprised of 74 % of Ce^IV and 26 % of Ce^III. The latter is found to be responsible for its reductive addition behavior towards CO, SO₂, and NO. This is the first report to date on the influence of the multiconfigurational ground state on the reactivity of a metal–oxo complex.