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Register a new userThe Moduli Space of smooth Ample Hypersurfaces in Abelian Varieties
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We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmuller space consisting of Kahler complex structures.
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We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.
We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.
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 consider the family $mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d geq 2$, and study the map $Phi_d:mathrm{MP}_dto widetilde{Lambda}_d subset mathbb{C}^d / mathfrak{S}_d$ which maps each $f in mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show that the local fiber structure of the map $Phi_d$ around $bar{lambda} in widetilde{Lambda}_d$ is completely determined by certain two sets $mathcal{I}(lambda)$ and $mathcal{K}(lambda)$ which are subsets of the power set of ${1,2,ldots,d }$. Moreover for any $bar{lambda} in widetilde{Lambda}_d$, we give an algorithm for counting the number of elements of each fiber $Phi_d^{-1}left(bar{lambda}right)$ only by using $mathcal{I}(lambda)$ and $mathcal{K}(lambda)$. It can be carried out in finitely many steps, and often by hand.
Faltings proved that there are finitely many abelian varieties of genus $g$ of a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Neron-Severi group of $A$, up to translation, as long as the dimension of $A$ is at least $4$. Our approach builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties.
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