1,721,053 research outputs found
On the Hodge conjecture for quasi-smooth intersections in toric varieties
Full text linkWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety PΣ2k+1 has Picard group Z. This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003)
Semistable Higgs bundles on elliptic surfaces
No full textWe analyze Higgs bundles (V, phi) on a class of elliptic surfaces pi : X -> B, whose underlying vector bundle V has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of V is reduced, then the Higgs field phi is vertical, while if the bundle V is fiberwise regular with reduced (respectively, integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of B and the degree of the fundamental line bundle of pi (respectively, if the fundamental line bundle is sufficiently ample), then phi is scalar. We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on the class of elliptic surfaces considered, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to X
Variational Calculus on Supermanifold and invariance properties of superspace field theories
No full textFiltrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau varieties
No full textWe consider Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced in [5; 4], and show that they have Jordan-Hölder filtrations whose quotients are stable, locally free and H-nflat. This is applied to show that curve semistable Higgs bundles on simply connected Calabi-Yau varieties have vanishing discriminant
Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles
No full textWe announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles.
A principal Higgs bundle on a compact K\"ahler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure
Normal bundles to Laufer curves in local Calabi-Yau threefolds
No full textWe compute the tree-level superpotential coming from B-branes wrapped on curves for a class of local Calabi--Yau threefolds, the Laufer (non rational) local curves. The superpotential is obtained by studying the deformation-obstruction map for the curve inside the threefold. It corresponds to the commutative limit of the brane potential. In this generality we claim that the noncommutative superpotential can be obtained from the commutative one in a simple way
On semistable principal bundles on complex projective manifolds
No full textLet be a simple linear algebraic group defined over
the field of complex numbers. Fix a proper parabolic subgroup
of , and also fix a nontrivial
antidominant character of . We prove that a holomorphic
principal --bundle over a
connected complex projective manifold is semistable
satisfying the condition that the second Chern class
vanishes
if and only if the line bundle over defined by
is numerically effective. Also, a
principal --bundle over
is semistable with if and only if for
every pair of the form
, where is a holomorphic map
to from a compact connected Riemann surface ,
and for every holomorphic reduction of structure group
to the subgroup ,
the line bundle over associated to
the principal --bundle for
is of nonnegative degree. Therefore,
is semistable with if and
only if for each pair of the above type the
--bundle over is semistable.
Similar results remain valid for principal
bundles over with a reductive linear
algebraic group as the
structure group. These generalize an earlier work of
Y. Miyaoka, \cite{Mi}, where he gave a characterization
of semistable vector bundles over a smooth projective curve.
Using these characterizations one can also
produce similar criteria for the semistability of parabolic
principal bundles over a compact Riemann surface
Mirror symmetry on K3 surfaces as a hyper-Kähler rotation
No full textWe show that under the hypotheses of Strominger, Yau and Zaslow, a mirror partner of a K3 surface X with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of X within its hyper-Kähler family of complex structures. Furthermore, the same hypotheses force the B-field to vanish
Semistability vs. nefness for (Higgs) vector bundles
No full textGeneralizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a
field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs , λs in the Grassmannians Grs(E) of
locally-free quotients of E and in the projective bundles PQs , respectively (here 0 < s <rkE and Qs is the universal quotient
bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability
is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the
nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class
Δ(E) = c2(E)− r−1
2r c1(E)2
Numerically flat Higgs vector bundles
No full textAfter providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish,
and that a Higgs bundle is Higgs-numerically flat if and only if it is has
a filtration whose quotients are flat stable Higgs bundles. We also study the relation between
these numerical properties of Higgs bundles and (semi)stability
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