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Register a new userOn boundedness of semistable sheaves
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Authors
Adrian Langer
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We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolovs inequality for semistable sheaves on a smooth projective variety of any dimension $ge 2$ without using any restriction theorems.
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An important classification problem in Algebraic Geometry deals with pairs $(E,phi)$, consisting of a torsion free sheaf $E$ and a non-trivial homomorphism $phicolon (E^{otimes a})^{oplus b}lradet(E)^{otimes c}otimes L$ on a polarized complex projective manifold $(X,O_X(1))$, the input data $a$, $b$, $c$, $L$ as well as the Hilbert polynomial of $E$ being fixed. The solution to the classification problem consists of a family of moduli spaces ${cal M}^delta:={cal M}^{delta-rm ss}_{a/b/c/L/P}$ for the $delta$-semistable objects, where $deltainQ[x]$ can be any positive polynomial of degree at most $dim X-1$. In this note we show that there are only finitely many distinct moduli spaces among the ${cal M}^delta$ and that they sit in a chain of GIT-flips. This property has been known and proved by ad hoc arguments in several special cases. In our paper, we apply refined information on the instability flag to solve this problem. This strategy is inspired by the fundamental paper of Ramanan and Ramanathan on the instability flag.
We bound the slope of sweeping curves in the fourgonal locus of the moduli space of genus g algebraic curves. Our results follow from some Bogomolov-type inequalities for weakly positive rank two vector bundles on ruled surfaces.
We introduce the notion of mixed-$omega$-sheaves and use it for the study of a relative version of Fujitas freeness conjecture.
It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when the general fibers of the corresponding Iitaka fibration are in a fixed bounded family of polarized Calabi-Yau pairs. As a consequence, we prove that in this case, the fibration is birationally bounded, and when it has terminal singularities, the corresponding minimal model is bounded in codimension 1.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the etale fundamental group of a curve. Faltings $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.
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