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Register a new userNumerical bounds for semi-stable families of curves or of certain higher dimensional manifolds
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Given an open subset U of a projective curve Y and a smooth family f:V-->U of curves, with semi-stable reduction over Y, we show that for a sub variation of Hodge structures of rank >2 the Arakelov inequality must be strict. For families of n-folds we prove a similar result under the assumption that the (n,0) component of the Higgs bundle defines fibrewise a birational map.
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This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura curve for g=3.
We prove the unirationality of the Ueno-type manifold $X_{4,6}$. $X_{4,6}$ is the minimal resolution of the quotient of the Cartesian product $E(6)^4$, where $E(6)$ is the equianharmonic elliptic curve, by the diagonal action of a cyclic group of order 6 (having a fixed point on each copy of $E(6)$). We collect also other results, and discuss several related open questions.
Let M be the moduli scheme of canonically polarized manifolds with Hilbert polynomial h. We construct for a given finite set I of natural numbers m>1 with h(m)>0 a projective compactification M of the reduced scheme underlying M such that the ample invertible sheaf L corresponding to the determinant of the direct image of the m-th power of the relative dualizing sheaf on the moduli stack, has a natural extension L to M. A similar result is shown for moduli of polarized minimal models of Kodaira dimension zero. In both cases natural means that the pullback of L to a curve C --> M, induced by a family f:X --> C is isomorphic to the determinant of the direct image of the m-th power of the relative dualizing sheaf whenever f is birational to a semi-stable family. Besides of the weak semistable reduction of Abramovich-Karu and the extension theorem of Gabber there are new tools, hopefully of interest by itself. In particular we will need a theorem on the flattening of multiplier sheaves in families, on their compatibility with pullbacks and on base change for their direct images, twisted by certain semiample sheaves. Following suggestions of a referee, we reorganized the article, we added several comments explaining the main line of the proof, and we changed notations a little bit.
We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN differentials is the set of zeroes of a twisted meromorphic RN differential, which we explicitly construct. Our main new tool is an explicit solution of the jump problem on Riemann surfaces in plumbing coordinates, by using the Cauchy kernel on the normalization of the nodal curve. Since this kernel does not depend on plumbing coordinates, we are able to approximate the RN differential on a smooth plumbed curve by a collection of meromorphic differentials on the irreducible components of a stable curve, with an explicit bound on the precision of such approximation. This allows us to also study these approximating differentials at suitable scales, so that the limit under degeneration is not identically zero. These methods can be applied more generally to study degenerations of differentials on Riemann surfaces satisfying various conditions.
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.
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