We prove a Darboux theorem for derived schemes with symplectic forms of degree k<0, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme X with symplectic form omega' of degree k is locally equivalent to (Spec A, omega) for Spec(A) an affine derived scheme in which the cdga A has Darboux-like coordinates with respect to which the symplectic form omega is standard, and in which the differential in A is given by a Poisson bracket with a Hamiltonian function Phi of degree k+1.
When k=-1, this implies that a -1-shifted symplectic derived scheme (X,omega') is Zariski locally equivalent to the derived critical locus Crit(Phi) of a regular function Phi: U --> A^1 on a smooth scheme U. We use this to show that the classical scheme t_0(X) has the structure of an algebraic d-critical locus, in the sense of Joyce.
In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.
The problem on the construction of antisymmetric paramodular forms of canonical weight $3$ was open since 1996. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to $(1,t)$-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight $3$ as Borcherds products whose first Fourier-Jacobi coefficient is a theta block.
We define an algebraic set in 23-dimensional projective space whose ℚ-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight 3examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight 2 is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over ℚ.
We construct a filtration on an integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each global Weyl module there is given by the corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type ADEADE in terms of Schubert subvarieties of thick affine Grassmanian, as predicted by Boris Feigin.
We prove that for l ≥ 3 the space of Bridgeland stability conditions on the Kronecker quiver with l parallel arrows is biholomorphic to C × H as a complex manifold.
We study two rational Fano threefolds with an action of the icosahedral group 𝔄5. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are 𝔄5-Fano varieties that are 𝔄5-birationally superrigid. This gives two new embeddings of the group 𝔄5 into the space Cremona group.
The EA-matrix integrals, introduced in Barannikov (Comptes Rendus Math 348:359–362, 2006), are studied in the case of graded associative algebras with odd or even scalar product. I prove that the EA-matrix integrals for associative algebras with scalar product are integrals of equivariantly closed differential forms with respect to the Lie algebra glN(A)glN(A).
In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the γ-th moments of gaps for all γ\leq 2.
In this paper, we prove that for any A>0 there exist infinitely many primes p for which sums of the Legendre symbols modulo p over an interval of length (ln p)^A can take large values.
We prove that a smooth well-formed Fano weighted complete intersection of codimension 2 has a nef partition. We discuss applications of this fact to Mirror Symmetry. In particular we list all nef partitions for smooth well-formed Fano weighted complete intersections of dimensions 4 and 5 and present weak Landau–Ginzburg models for them.
We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on P1bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi–Yau varieties.
We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.