A remark on Ulrich and ACM bundles

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.

The geometry ofN=2 supergravity is related to the variations of Hodge structure for “formal” Calabi-Yau spaces. All known results in this branch of algebraic geometry are easily recovered from supersymmetry arguments. This identification has a physical meaning for a type IIB superstring compactified on a Calabi-Yau 3-fold. We give exact (non-perturbative) results for the string effective lagrangian. Our geometrical framework suggests a re-formulation of the Gepner conjecture about (2,2) superconformal theories as the solution to theSchottky problem for algebraic complex manifolds having trivial canonical bundle.

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103–134], Lu and Tian [Complex structures on connected sums of S3 × S3. In: Manifolds and Geometry, Pisa, 1993, 284–293] who used only quintics in ℙ4.

We prove that smooth, projective, K-trivial, weakly ordinary varieties over a perfect field of characteristic p>0 are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer’s results, implies that varieties of the above type have strongly semistable tangent bundles with respect to every polarization.

We study the deformation theory of Calabi-Yau structures in strongly pseudo-convex manifolds with trivial canonical bundles. Our approach could be considered as an alternative proof for a theorem of H. Laufer on the deformation of strongly pseudo-convex surfaces.

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds $M$ with the trivial canonical bundle on the universal cover $\widetilde{M}$.

Let ( X , ω ) (X,\omega ) be a compact hermitian manifold of dimension n n . We study the asymptotic behavior of Monge-Ampère volumes ∫ X ( ω + d d c φ ) n \int _X (\omega +dd^c \varphi )^n , when ω + d d c φ \omega +dd^c \varphi varies in the set of hermitian forms that are d d c dd^c -cohomologous to ω \omega . We show that these Monge-Ampère volumes are uniformly bounded if ω \omega is “strongly pluripositive”, and that they are uniformly positive if ω \omega is “strongly plurinegative”. This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at 6 6 -dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study 6 6 -dimensional solvmanifolds with trivial canonical bundle.

We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact K\"ahlerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose critical locus is compact but need not consist of isolated points. We also show how this construction specializes to the case when $X$ is Stein and $W$ has finite critical set, in which case one recovers a simpler mathematical model.

Surfaces of minimal degree of tame representation type and mutations of Cohen–Macaulay modules

Let [Formula: see text] be a surface with [Formula: see text] and endowed with a very ample line bundle [Formula: see text] such that [Formula: see text]. We show that [Formula: see text] supports special (often stable) Ulrich bundles of rank [Formula: see text], extending a recent result by A. Beauville. Moreover, we show that such an [Formula: see text] supports families of dimension [Formula: see text] of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large [Formula: see text] except for very few cases. We also show that the same is true for each linearly normal non-special surface with [Formula: see text] in [Formula: see text] of degree at least [Formula: see text], Enriques surface and anticanonical rational surface.

Ulrich bundles on ruled surfaces

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.

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

We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface tilde{B} that is related through toric geometry to the line bundle −6KB. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension.

Rank 2 Ulrich bundles on general double plane covers