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Register a new userCohomological rigidity for toric Fano manifolds of small dimensions or large Picard numbers
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The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID numbers are 50 and 57 on a list of {O}bro) have isomorphic cohomology rings and their total Pontryagin classes are preserved under an isomorphism between their cohomology rings, but we do not know whether they are diffeomorphic or homeomorphic.
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In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds $X$ and $X$, the following conditions are equivalent: (1) the upper triangular matrix associated to $X$ can be transformed into that of $X$ by those three operations; (2) $X$ and $X$ are diffeomorphic; (3) the integral cohomology rings of $X$ and $X$ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.
It is proved that if two quasitoric manifolds of dimension $le 2p^2-4$ for a prime $p$ have isomorphic cohomology rings, then they have the same $p$-local stable homotopy type.
A quasitoric manifold is a smooth manifold with a locally standard torus action for which the orbit space is identified with a simple polytope. For a class of topological spaces, the class is called strongly cohomologically rigid if any isomorphism of cohomology rings can be realized as a homeomorphism. This paper shows the strong cohomological rigidity of the class of quasitoric manifolds over $I^3$.
We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal to the rank of G (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper we show that the rational homogeneous space G/P is rigid under Fano deformation, providing that G is a simple algebraic group of type ADE, the Picard number equal to rank(G)-1 and G/P is not biholomorphic to F(1, 2, P^3) or F(1, 2, Q^6). The variety F(1, 2, P^3) is the set of flags of projective lines and planes in P^3, and F(1, 2, Q^6) is the set of flags of projective lines and planes in 6-dimensional smooth quadric hypersurface. We show that F(1, 2, P^3) have a unique Fano degeneration, which is explicitly constructed. The structure of possible Fano degeneration of F(1, 2, Q^6) is also described explicitly. To prove our rigidity result, we firstly show that the Fano deformation rigidity of a homogeneous space of type ADE can be implied by that property of suitable homogeneous submanifolds. Then we complete the proof via the study of Fano deformation rigidity of rational homogeneous spaces of small Picard numbers. As a byproduct, we also show the Fano deformation rigidity of other manifolds such as F(0, 1, ..., k_1, k_2, k_2+1, ..., n-1, P^n) and F(0, 1, ..., k_1, k_2, k_2+1, ..., n, Q^{2n+2}) with 0 <= k_1 < k_2 <= n-1.
The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank $1$ or $2$. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.
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