Algebraic integrability of foliations with numerically trivial canonical bundle

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation. Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class. In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau-- and irreducible holomorphic-symplectic manifolds, respectively.

Pathologies of the Hilbert scheme of points of a supersingular Enriques surface

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 prove that a geometrically integral smooth projective 3‐fold with nef anti‐canonical class and negative Kodaira dimension over a finite field of characteristic and cardinality has a rational point. Additionally, under the same assumptions on and , we show that a smooth projective 3‐fold with trivial canonical class and non‐zero first Betti number has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi–Yau 3‐fold pairs over perfect fields.

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface $X$ with at most double curves. We prove that we still have a family of smoothings of $X$ in a weak sense even when $X$ is not Kählerian or $H^{1}(X,\\mathcal{O}_{X})\\neq 0$ , in which cases the smoothability result of Friedman [Fr] is not applicable.

We study positivity properties of the moduli (b-)divisor associated to a relative log pair (X, B)/Y with relatively trivial log canonical class.

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 study Le Potier's strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank 2 sheaves with trivial first Chern class and second Chern class 2, and the moduli space of 1-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $Σ_e:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))$ except for $e=1$. When $e=1$, our result applies to $L=aG+bF$ with $b\geq a+[a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2=-1$ and $[a/2]$ is the integral part of $a/2$.

Strange duality on [formula omitted] via quiver representations

A remark on Ulrich and ACM bundles

Constraints on the Chern classes of a vector bundle on a possibly singular algebraic variety are found, which are stronger than the obvious Hodge theoretic constraints. This is done by showing that these lift to Chern classes in the hypercohomology of the complex of Kähler differentials.

Let <TEX>$M_c$</TEX> = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes <TEX>$c_1=0\;and\;c_2=c$</TEX> on a K3 surface X, where O(1) is a generic ample line bundle on X. When <TEX>$c=2n\geq4$</TEX> is even, <TEX>$M_c$</TEX> is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, <TEX>$M_c$</TEX> has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of <TEX>$M_{2n}$</TEX> for <TEX>$n\geq3$</TEX>. In this paper, we show that there is no crepant resolution of <TEX>$M_{2n}$</TEX> for <TEX>$n\geq3$</TEX>. This obviously implies that there is no symplectic desingularization.

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch‐Riemann‐Roch theorem. As important special cases, we obtain new (or recover well‐known) formulae for the Baum‐Fulton‐MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom‐Milnor L‐classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus‐orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus‐invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice‐point counting in lattice polytopes and their subcomplexes, yielding generalized Pick‐type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz‐Riemann‐Roch theorem of Edidin‐Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L‐classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Let E be a semistable, rank two vector bundle of degree d on a Riemann surface C of genus g > 1, i.e. such that the minimal degree s of a tensor product of E with a line bundle having a nonzero section is nonnegative.We give an analogue of Clifford's lemma by showing that E has at most (d-s)/2 + 6 independent sections, where 6 is 2 or 1 according to whether the Krawtchouk polynomial Kr(n, N) is zero or not at r = (d-s)/2 + 1, n = g, N = 2g-s (the analogous bound for nonsemistable rank two bundles being stronger but easier to prove).This gives an answer to the problem posed by Severi asking for the minimal degree of a directrix of a ruled surface.In some cases, namely if s has maximal value s = g, or if s > gonality(C) -2, or if E is general among those of the same Segre invariant s, or also if the genus is a power of two, we prove the bound holds with 3 = 1.The theory of Krawtchouk polynomials investigates which triples (g, s, d) provide zeros of Kr(n,N).Then, they generate invariants which one may expect to be associated to a Severi bundle, i.e., to a rank two semistable bundle reaching the bound with 3 = 2.According to this theory, there are only a finite number of such triples (g, s, d) for each value of ds, with the exception that there are infinitely many triples with d-s = 2 or 4. We then find all the Severi bundles corresponding to those two exceptional values of ds.We thank David Butler for a personal communication which we have used in proving Proposition 3.1.We also thank Enrique Arrondo, Rafael Hernandez, Jorge Jimenez and Bernardo L6pez for conversations during the preparation of this article.JAVIER CILLERUELO AND IGNACIO SOLS having a nonzero section or, equivalently, the integer s such that the maximal degree of a line subbundle L c-E is d2S.In [16] Nagata proved that such an integer s is at most g and that we can assume s > 2-2g, because otherwise H1(L* ® (E/L)) = O, which implies that E is decomposable and the rank one Clifford bound is enough to bound its sections.Let us recall, furthermore, that the bundle E is stable if s > 0, and semistable if s > 0. If E is special, i.e., if h°(E) is neither zero nor given by the Riemann-Roch formula h°(E) = d + 2 -2g, then we have s < d < 4g -4s.Therefore we center our study in this "Severi range" of values of d.This study is trivial in case E is not semistable, providing then a sharp bound (stronger than the "Severi bound" for semistable bundles in Theorem 0.2 below): PROPOSITION 0.1.Let E be a rank two bundle of Segre invariant s, 2 -2g < s < 0, and degree d on a Riemann surface of genus g.If s < d < -s then h°(E) < [ds] + 1. d + 3 if d is odd If-s < d < 4g-4+s then h(E) < + 2 if d is even, and d s (4) + 1 if d is even, and d g s (4).If4g-4 + s < d < 4g s then hO(E) < [3d] g + 2. Furthermore, for each g, s, d in this range, there are a Riemann surface of genus g, and a rank two bundle of invariant s and degree d reaching the bound.Indeed, by Serre duality it is enough to show the bound in the case d < 2g -2.Let L be a subbundle of maximal degree 0 - L - E - L' -0, i.e. degL = d-2 < 2g-2 and degL' = d-< g -2.Then the desired bound follows from h°(L) < [d] + 1 and h°(L') < max{0, [d+] + 1} (for the case -s< d < 2g-2, we note that d+s _ d-s or d+sd-s (mod 4) depending on whether d is even or odd).For the second assertion, since our assumption on the invariant s implies that g > 2, we can take C to be a hyperelliptic curve, and E = L L', where degL = d, degL' = d+s and h°(L) [ds] + 1, h°(L') = max{0, [d+s] + 1 In Section 1 we present the proof of our main theorem, in which the binary Krawtchouk polynomials Kr(n,N) appear.Let us recall that these polynomials are defined, for each r > 0, by their generating function Kr(n, N)zr = (1z)n(l + z)N-n r

For a reflexive sheaf E, we obtain an upper bound, in terms of the Chern classes of E , for the vanishing of the groups H 1(P 3 E(k)).We also show that, for every c 2 ≥ 2, there exists a normalized rank 2 stable reflexive sheaf over P 3 for which the bounds given are the best possible.