Strange duality on rational surfaces

Strange duality on [formula omitted] via quiver representations

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi–Yau threefold \(K_S\) are described by semi-stable coherent sheaves on S. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson–Thomas invariants) coincide with the Vafa–Witten invariants of S (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential (Q, W) constructed from the exceptional collection. We spell out the dictionary between the Chern class \(\gamma \) and polarization J on S versus the dimension vector \(\vec {N}\) and stability parameters \(\vec {\zeta }\) on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices \(\Omega _\star (\gamma )\) at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa–Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that (1) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and (2) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi–Yau geometries.

Given a family of nonsingular complex projective surfaces, there is a corresponding family of Hilbert schemes of zero dimensional subschemes. We prove that the Chern classes, with values in the rational Chow groups, of the de Rham bundles for such a family of Hilbert schemes vanish. A similar result is proved for any relative moduli space of rank one sheaves with trivial integral first Chern class over any family of complex projective surfaces.

We show how the “finite Quot scheme method” applied to Le Potier’s strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the nth Catalan number.

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.

We give an algorithm for computing the contact homology of some Brieskorn manifolds. As an application, we construct infinitely many contact structures on the class of simply connected contact manifolds that admit nice contact forms (i.e. no Reeb orbits of degree -1,0 or 1) and have index positivity with trivial first Chern class.

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

We describe various properties of Hirzebruch surfaces and related constructions: degenerations, braid monodromy, Galois covers and their Chern classes.

We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its C 0 -closure Ham ¯ ( M , ω ) in the set of all homeomorphisms, and its completion with respect to the spectral norm Ham ^ ( M , ω ) . We prove that in some situations, namely complex projective spaces and rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be non-trivial in π 1 ( Ham ( M , ω ) ) remain non-trivial in π 1 ( Ham ^ ( M , ω ) ) . This yields in particular cases, including ℂ P 2 and the monotone S 2 × S 2 , the injectivity of the map π 1 ( Ham ( M , ω ) ) → π 1 ( Ham ^ ( M , ω ) ) induced by the inclusion. The same results hold for the Hofer completion of Ham ( M , ω ) . Moreover, whenever the spectral norm is known to be C 0 -continuous, they also hold for Ham ¯ ( M , ω ) . Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on π 1 ( Ham ( M , ω ) ) . Some of these computations were known before, but we also present new ones which might be of independent interest. For example, we show that the spectral pseudo-norm is degenerate when ( M , ω ) is any non-monotone S 2 × S 2 . At the contrary, it is a genuine norm when M is the 1-point blow-up of ℂ P 2 ; it is unbounded for small sizes of the blow-up and become bounded starting at the monotone one.

We consider the embedding capacity functions \(c_{H_{b}}(z)\) for symplectic embeddings of ellipsoids of eccentricity z into the family of nontrivial rational Hirzebruch surfaces Hb with symplectic form parametrized by b ∈ [0, 1). This function was known to have an infinite staircase in the monotone cases (b = 0 and b = 1∕3). It is also known that for each b there is at most one value of z that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked b-intervals, consisting of b-parameters where the embedding capacity function for Hb does not contain an infinite staircase. There is one sequence in each of the intervals (0, 1∕5), (1∕5, 1∕3), and (1∕3, 1). We then establish six sequences of associated infinite staircases, one occurring at each endpoint of the blocked b-intervals. The staircase numerics are variants of those in the Fibonacci staircase for the projective plane (the case b = 0). We also show that there is no staircase at the point b = 1∕5, even though this value is not blocked. The focus of this paper is to develop techniques, both graphical and numeric, that allow identification of potential staircases, and then to understand the obstructions well enough to prove that the purported staircases really do have the required properties. A subsequent paper will explore in more depth the set of b that admit infinite staircases.

In this paper we study Zappatic surfaces. We give a combinatorial description of whether the Galois cover of degeneration of a surface to a union of planes is simply connected. Additionally, we compute Chern numbers and the signatures of the Galois covers of the surfaces and present applications to Hirzebruch and Veronese surfaces.

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a non trivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the Gromov-Witten invariants, which turn out to be very symmetric.

We prove that up to birational equivalence, there exists only a finite number of families of Calabi-Yau threefolds (i.e. a threefold with trivial canonical class and factorial terminal singularities) which have an elliptic fibration to a rational surface. This strengthens a result of B. Hunt that there are only a finite number of possible Euler characteristics for such threefolds.

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if $X$ is a smooth affine scheme of dimension $d$ over a field $k$ of finite 2-cohomological dimension (with $\mathrm{char}(k)\neq 2)$ and $E$ is a rank $d$ vector bundle over $X$, vanishing of the Chow-Witt theoretic Euler class of $E$ is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when $k$ is of small 2-cohomological dimension.

% We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\p3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\sG$ is a subfoliation of a codimension one distribution $\sF$ with isolated singularities, then $\sing(\sG)$ is a curve. As a consequence, we give a criterion to decide whether $\sG$ is globally given as the intersection of $\sF$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.