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We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.
Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of
generic hyperplanes in
, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of
generic hyperplanes in
-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety
. By localizing, we deduce that the (fully) wrapped Fukaya category of the
-dimensional pair of pants is equivalent to the derived category of
. We also prove similar equivalences for finite abelian covers of the
-dimensional pair of pants.
We compute the
B-model Gromov–Witten invariant of an elliptic curve
directly from the derived category
. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic
described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
We prove some numerical inequality for the Horikawa indices for Eisenbud–Harris special nonhyperelliptic fibrations of genus 4 on algebraic surfaces under the assumption that the multiplication map of the fibration is not surjective. Furthermore, we prove that the inequality is best possible by constructing the examples satisfying the equality.
We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.
We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.
We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.
We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.
In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.
For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.
We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a
surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all
-dimensional families of manifolds of
-type with a non-symplectic involution for
and provide examples arising as moduli spaces of twisted sheaves on a
This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.
We prove the integral Hodge conjecture for all 3-folds
of Kodaira dimension zero with
not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.
We study the variations of mixed Hodge structures (VMHS) associated with a pencil
of equisingular hypersurfaces of degree
with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family
. The notion of a set of singular points being in homologically good position is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths’ description of the
-term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil
of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention.
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over
is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by
Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using
-deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.
We compute the Hodge ideals of
-divisors in terms of the
-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.
We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index
, then the degree of irrationality of a very general complex Fano hypersurface of index
and dimension n is bounded from below by a constant times
. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic
argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.
We prove the Lipman–Zariski conjecture for complex surface singularities with
is the geometric genus,
is the sum of the genera of exceptional curves and
is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.
In this note, using methods introduced by Hacon et al. [‘Boundedness of varieties of log general type’, Proceedings of Symposia in Pure Mathematics, Volume 97 (American Mathematical Society, Providence, RI, 2018) 309–348], we study the accumulation points of volumes of varieties of log general type. First, we show that if the set of boundary coefficients Λ satisfies the descending chain condition (DCC), is closed under limits and contains 1, then the corresponding set of volumes satisfies the DCC and is closed under limits. Then, we consider the case of ε-log canonical varieties, for 0 < ε < 1. In this situation, we prove that if Λ is finite, then the corresponding set of volumes is discrete.