The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of $\mathbb{C}^*$ whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant $\mathrm{O}_2(\mathbb{C})$-vector bundles.
Consider the ten-dimensional spinor variety in the projectivization of a
half-spin representation of dimension sixteen. The intersection X of two
general translates of this variety is a smooth Calabi-Yau fivefold, as well as
the intersection Y of their projective duals. We prove that although X and Y
are not birationally equivalent, they are derived equivalent and L-equivalent
in the sense of Kuznetsov and Shinder.
Using etale cohomology, we define a birational invariant for varieties in
characteristic $p$ that serves as an obstruction to uniruledness - a variant on
an obstruction to unirationality due to Ekedahl. We apply this to
$\overline{M}_{1,n}$ and show that $\overline{M}_{1,n}$ is not uniruled in
characteristic $p$ as long as $n \geq p \geq 11$. To do this, we use Deligne's
description of the etale cohomology of $\overline{M}_{1,n}$ and apply the
theory of congruences between modular forms.
We study some aspects of the $\lambda_g$ pairing on the tautological ring of
$M_g^c$, the moduli space of genus $g$ stable curves of compact type. We
consider pairing kappa classes with pure boundary strata, all tautological
classes supported on the boundary, or the full tautological ring. We prove that
the rank of this restricted pairing is equal in the first two cases and has an
explicit formula in terms of partitions, while in the last case the rank
increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on
the tautological ring of $M_g$.
We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \to \text{Lie}(G, I) \to E \to G \to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.
In this note we show that any lattice in a simple p-adic Lie group is not the
fundamental group of a compact Ka\"hler manifold, as well as some variants of
this result.
When $W$ is a finite Coxeter group acting by its reflection representation on
$E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C},
{\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb
C}$, smooth with respect to the stratification by reflection hyperplanes. By
using Kapranov and Schechtman's recent analysis of perverse sheaves on
hyperplane arrangements, we find an equivalence of categories from
${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of
finite-dimensional modules over an algebra given by explicit generators and
relations.
We also define categories of equivariant perverse sheaves on affine
buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building
of a $p$-adic group $G$. In this setting, we find that a construction of
Schneider and Stuhler gives equivariant perverse sheaves associated to depth
zero representations.
We show that if a Fano manifold does not admit Kahler-Einstein metrics then
the Kahler potentials along the continuity method subconverge to a function
with analytic singularities along a subvariety which solves the homogeneous
complex Monge-Ampere equation on its complement, confirming an expectation of
Tian-Yau.
We classify singular Enriques surfaces in characteristic two supporting a
rank nine configuration of smooth rational curves. They come in one-dimensional
families defined over the prime field, paralleling the situation in other
characteristics, but featuring novel aspects. Contracting the given rational
curves, one can derive algebraic surfaces with isolated ADE-singularities and
trivial canonical bundle whose Q_l-cohomology equals that of a projective
plane. Similar existence results are developed for classical Enriques surfaces.
We also work out an application to integral models of Enriques surfaces (and K3
surfaces).
To reinforce the analogy between the mapping class group and the Cremona
group of rank $2$ over an algebraic closed field, we look for a graph
analoguous to the curve graph and such that the Cremona group acts on it
non-trivially. A candidate is a graph introduced by D. Wright. However, we
demonstrate that it is not Gromov-hyperbolic. This answers a question of A.
Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono\"i
tesselation of the Cremona group introduced in a previous work of the autor. We
show that one is quasi-isometric to the Wright graph. We prove that the second
one is Gromov-hyperbolic.
The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the
moduli part in a canonical bundle formula is semiample on a birational
modification. We prove that the restriction of the moduli part to any
sufficiently high divisorial valuation is semiample, assuming the conjecture in
lower dimensions.
In this paper we study rational surface singularities R with star shaped dual
graphs, and under very mild assumptions on the self-intersection numbers we
give an explicit description of all their special Cohen-Macaulay modules. We do
this by realising R as a certain Z-graded Veronese subring S^x of the
homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X,
and we realise the special CM modules as explicitly described summands of the
canonical tilting bundle on X. We then give a second proof that these are
special CM modules by comparing qgr S^x and coh X, and we also give a necessary
and sufficient combinatorial criterion for these to be equivalent categories.
In turn, we show that qgr S^x is equivalent to qgr of the reconstruction
algebra, and that the degree zero piece of the reconstruction algebra coincides
with Ringel's canonical algebra. This implies that the reconstruction algebra
contains the canonical algebra, and furthermore its qgr category is derived
equivalent to the canonical algebra, thus linking the reconstruction algebra of
rational surface singularities to the canonical algebra of representation
theory.
We prove that the $\ell$-adic Chern classes of canonical extensions of
automorphic vector bundles, over toroidal compactifications of Shimura
varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the
$\ell$-adic cohomology of the minimal compactifications. These are invariant
under the Galois group of the $p$-adic field above which the variety and the
bundle are defined.
There are two ways to define the Swan conductor of an abelian character of
the absolute Galois group of a complete discrete valuation field. We prove that
these two Swan conductors coincide.
In this article, we study isomorphisms between complements of irreducible
curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically
closed field. Of particular interest are rational unicuspidal curves. We prove
that if there exists a line that intersects a unicuspidal curve $C \subset
\mathbb{P}^2$ only in its singular point, then any other curve whose complement
is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to
$C$. This generalizes a result of H. Yoshihara who proved this result over the
complex numbers. Moreover, we study properties of multiplicity sequences of
irreducible curves that imply that any isomorphism between the complements of
these curves extends to an automorphism of $\mathbb{P}^2$. Using these results,
we show that two irreducible curves of degree $\leq 7$ have isomorphic
complements if and only if they are projectively equivalent. Finally, we
describe new examples of irreducible projectively non-equivalent curves of
degree $8$ that have isomorphic complements.
We consider the moduli space of stable torsion free sheaves of any rank on a
smooth projective threefold. The singularity set of a torsion free sheaf is the
locus where the sheaf is not locally free. On a threefold it has dimension
$\leq 1$. We consider the open subset of moduli space consisting of sheaves
with empty or 0-dimensional singularity set.
For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the
generating function of topological Euler characteristics of these open subsets
equals a power of the MacMahon function times a Laurent polynomial. This
Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon
replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open
subsets equal the entire moduli space.
The proof involves wall-crossing from Quot schemes of a higher rank reflexive
sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret
this sublocus in terms of the singularities of the reflexive sheaf.
We study the singularities of Legendrian subvarieties of contact manifolds in
the complex-analytic category and prove two rigidity results. The first one is
that Legendrian singularities with reduced tangent cones are
contactomorphically biholomorphic to their tangent cones. This result is partly
motivated by a problem on Fano contact manifolds. The second result is the
deformation-rigidity of normal Legendrian singularities, meaning that any
holomorphic family of normal Legendrian singularities is trivial, up to
contactomorphic biholomorphisms of germs. Both results are proved by exploiting
the relation between infinitesimal contactomorphisms and holomorphic sections
of the natural line bundle on the contact manifold.
We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures.
Let k be an uncountable algebraically closed field and let Y be a smooth
projective k-variety which does not admit a decomposition of the diagonal. We
prove that Y is not stably birational to a very general hypersurface of any
given degree and dimension. We use this to study the variation of the stable
birational types of Fano hypersurfaces over fields of arbitrary characteristic.
This had been initiated by Shinder, whose method works in characteristic zero.