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For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.
By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-
surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-
surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to
-lattices. In particular, for the
-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-
surfaces. The analogous
-dimensional case, which corresponds to hyperelliptic degree-
surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.
The notion of Hochschild cochains induces an assignment from
, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor
, where the latter denotes the category of monoidal DG categories and bimodules. Any functor
gives rise, by taking modules, to a theory of sheaves of categories
. In this paper, we study
. Loosely speaking, this theory categorifies the theory of
-modules, in the same way as Gaitsgory’s original
categorifies the theory of quasi-coherent sheaves. We develop the functoriality of
, its descent properties and the notion of
-affineness. We then prove the
-affineness of algebraic stacks: for
a stack satisfying some mild conditions, the
is equivalent to the
-category of modules for
, the monoidal DG category of higher differential operators. The main consequence, for
quasi-smooth, is the following: if
is a DG category acted on by
admits a theory of singular support in
is the space of singularities of
. As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of
, thereby equipping objects of
with singular support in
We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.
In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic
. Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic
is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.
We prove that a reduced and irreducible algebraic surface in
containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.
We extend the Kuga–Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga–Satake abelian varieties associated with polarized variations of K3 type Hodge structures over the punctured disc.
Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index
in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra
is formal for any sheaf
polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.
In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.
We study the action of the inertia operator on the motivic Hall algebra and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.
By use of a natural extension map and a power series method, we obtain a local stability theorem for
-Kähler structures with the
-lemma under small differentiable deformations.
We describe all degenerations of three-dimensional anticommutative algebras
and of three-dimensional Leibniz algebras
. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.
We analyse infinitesimal deformations of pairs
a coherent sheaf on a smooth projective variety
over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.
We compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.
be a smooth projective curve of genus
over an algebraically closed field
. We show that for any integers
, there exists a maximally Frobenius destabilised stable vector bundle of rank
if and only if
be the irreducible Hermitian symmetric domain of type
. There exists a canonical Hermitian variation of real Hodge structure
of Calabi–Yau type over
. This short note concerns the problem of giving motivic realizations for
. Namely, we specify a descent of
and ask whether the
can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When
, we give a motivic realization for
, we show that the unique irreducible factor of Calabi–Yau type in
can be realized motivically.
This paper contains two results on Hodge loci in
. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in
. It is proved that the image under the period map of a divisor in
is not contained in a proper totally geodesic subvariety of
. It follows that a Hodge locus in
has codimension at least 2.
We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.