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Register a new userOn the cohomology of regular surfaces isogenous to a product of curves with $chi(mathcal{O}_S)=2$
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Authors
Matteo A. Bonfanti
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Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of unmixed type with $chi (mathcal{O}_S)=2$. In particular we found two families of surfaces of general type with maximal Picard number.
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A complex surface $S$ is said to be isogenous to a product if $S$ is a quotient $S=(C_1 times C_2)/G$ where the $C_i$s are curves of genus at least two, and $G$ is a finite group acting freely on $C_1 times C_2$. In this paper we classify all regular surfaces isogenous to a product with $chi(mathcal O_S) = 2$ under the assumption that the action of $G$ is unmixed i.e. no element of $G$ exchange the factors of the product $C_1 times C_2$.
We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has image $Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for $p_g=6$. And a connected component of the moduli space consisting of surfaces $S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $geq 2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$ with $K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP s, i.e. surfaces $S$ isogenous to a product of curves $(C_1 times C_2 )/ G$; in our examples the group $G$ is elementary abelian, $G = (mathbb{Z}/m)^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory.
In this paper we study emph{threefolds isogenous to a product of mixed type} i.e. quotients of a product of three compact Riemann surfaces $C_i$ of genus at least two by the action of a finite group $G$, which is free, but not diagonal. In particular, we are interested in the systematic construction and classification of these varieties. Our main result is the full classification of threefolds isogenous to a product of mixed type with $chi(mathcal O_X)=-1$ assuming that any automorphism in $G$, which restricts to the trivial element in $Aut(C_i)$ for some $C_i$, is the identity on the product. Since the holomorphic Euler-Poincare-characteristic of a smooth threefold of general type with ample canonical class is always negative, these examples lie on the boundary, in the sense of threefold geography. To achieve our result we use techniques from computational group theory. Indeed, we develop a MAGMA algorithm to classify these threefolds for any given value of $chi(mathcal O_X)$.
In this paper, we classify the minimal surfaces of general type with $chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32 respectively. Among them, the family of surfaces having a genus 2 fibration forms an irreducible component of $mathfrak{M}_{chi=5, K^{2}=9}$.
Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{mathbb{C}})/B$ such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in $GL(n-1,{mathbb{C}})/B$, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincare polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of Hessenberg Schubert polynomials in the context of regular nilpotent Hessenberg varieties, and outline several open questions pertaining to them.
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