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We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.
We prove that the maximal number of conics in a smooth sextic
$K3$
-surface
$X\subset \mathbb {P}^4$
is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the
$(n-1,n)$
th mild
$\partial \overline {\partial }$
-lemma by power series method and the other one on p-Kähler structures with the deformation invariance of
$(p,p)$
-Bott–Chern numbers.
Let q a prime power and
${\mathbb F}_q$
the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in
${\mathbb F}_q((T^{-1}))$
. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.
Let G be a permutation group on a set
$\Omega $
of size t. We say that
$\Lambda \subseteq \Omega $
is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of
$\Lambda $
. We define the height of G to be the maximum size of an independent set, and we denote this quantity
$\textrm{H}(G)$
. In this paper, we study
$\textrm{H}(G)$
for the case when G is primitive. Our main result asserts that either
$\textrm{H}(G)< 9\log t$
or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study
$\textrm{I}(G)$
, the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either
$\textrm{I}(G)<7\log t$
or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category
$\mathsf {D}^{\mathsf f}(R)$
, which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in
$\mathsf {D}^{\mathsf f}(R)$
is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings.
For a finite-dimensional Lie algebra
$\mathfrak {L}$
over
$\mathbb {C}$
with a fixed Levi decomposition
$\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$
, where
$\mathfrak {g}$
is semisimple, we investigate
$\mathfrak {L}$
-modules which decompose, as
$\mathfrak {g}$
-modules, into a direct sum of simple finite-dimensional
$\mathfrak {g}$
-modules with finite multiplicities. We call such modules
$\mathfrak {g}$
-Harish-Chandra modules. We give a complete classification of simple
$\mathfrak {g}$
-Harish-Chandra modules for the Takiff Lie algebra associated to
$\mathfrak {g} = \mathfrak {sl}_2$
, and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple
$\mathfrak {g}$
-Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff
$\mathfrak {sl}_2$
and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple
$\mathfrak {g}$
-Harish-Chandra modules.