Some definitions of finiteness: Corrigenda

1 Some definitions 13

.1.2 Some Definitions

The feasible region of this LP is in general, a convex polyhedron. Visualize it as a polygon in 2 dimensions, for simplicity. Now, maximizing yT b is the same as maximizing the projection of the vector y in the direction represented by vector b. For whichever direction b we choose, the point y that maximizes yT b cannot lie strictly in the interior of the feasible region. The reason is that, fr...

Some finiteness results in the category U

This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups.

Some finiteness results for Calabi–Yau threefolds

We investigate the moduli theory of Calabi–Yau threefolds, and using Griffiths’ work on the period map, we derive some finiteness results. In particular, we confirm a prediction of Morrison’s Cone Conjecture.

Some Finiteness Theorems for Abelian Varieties

Last time we saw (see Proposition 6.5 in those lecture notes) that an abelian variety A of dimension g over K, the fraction field of a henselian dvr R, acquires semistable reduction over K(A[`]) for ` equal to 4 or an odd prime, with ` not divisible by the residue characteristic. (The same circle of ideas yielded the non-obvious fact that over K the intersection of two “semistable field extensi...