On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces

We give an explicit procedure which computes for degree $d \leq 3$ the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface $X$ as homogeneous polynomials of degree $d$ in the correlation functions of degree 1 (number of lines). We extend this formalism to the case of Calabi-Yau hypersurfaces and explain how the polynomial property is preserved. Our key tool is the construction of universal recursive formulas which express the structural constants of the quantum cohomology ring of $X$ as weighted homogeneous polynomial functions in the constants of the Fano hypersurface with the same degree and dimension one more. We propose some conjectures about the existence and the form of the recursive formulas for the structural constants of rational curves of arbitrary degree. Our recursive formulas should yield the coefficients of the hypergeometric series used in the mirror calculation. Assuming the validity of the conjectures we find the recursive laws for rational curves of degree 4 and 5.Comment: 32 pages, changed fonts, exact results on quintic rational curves are added. To appear in Commun. Math. Phy

The Fano normal function

The Fano surface $F$ of lines in the cubic threefold $V$ is naturally embedded in the intermediate Jacobian $J(V)$, we call 'Fano cycle' the difference $F-F^{-}$, this is homologous to 0 in $J(V)$. We study the normal function on the moduli space which computes the Abel-Jacobi image of the Fano cycle. By means of the related infinitesimal invariant we can prove that the primitive part of the normal function is not of torsion. As a consequence we get that, for a general $V, F-F^{-}$is not algebraically equivalent to zero in $J(V)$ (proved also by van der Geer and Kouvidakis (2010) [15] with different methods) and, moreover, that there is no divisor in $J V$ containing both $F$ and $F^{-}$and such that these surfaces are homologically equivalent in the divisor. Our study of the infinitesimal variation of Hodge structure for $V$ produces intrinsically a threefold $\Xi(V)$ in the Grassmannian of lines $\mathbb{G}$ in $\mathbb{P}^4$. We show that the infinitesimal invariant at $V$ attached to the normal function gives a section of a natural bundle on $\Xi(V)$ and more specifically that this section vanishes exactly on $\Xi \cap F$, which turns out to be the curve in $F$ parameterizing the 'double lines' in the threefold. We prove that this curve reconstructs $V$ and hence we get a Torelli-like result: the infinitesimal invariant for the Fano cycle determines $V$

CHARACTERIZATION OF GENETIC AND EPIGENETIC MODIFICATIONS IN A MODEL OF INFLAMMATION-DRIVEN CANCER

Chronic inflammation is causally associated to many types of tumor, and has been recently acknowledged as a cancer hallmark. Nevertheless, whether inflammation has an intrinsic mutagenic potential is still not directly proven or understood from a mechanistic point of view. Furthermore, it is as yet unclear whether inflammation could induce epigenetic modifications, and if these changes are relevant to tumor generation. Therefore, the aim of this work was to assess inflammation-derived genomic and epigenomic modifications at multiple stages of tumorigenesis in Mdr2-knockout mice, a model of purely inflammatory hepatocellular carcinoma (HCC). By ChIPseq profiling of H3K27Ac mark we reported the establishment of an inflammatory program specifically in hepatocytes starting from the pre-malignant, chronic inflammatory step. This inflammatory signature is retained up to the more advanced HCC stage, and is accompanied by the activation of members of the AP1 transcription factor family. In parallel, by whole exome sequencing, we observed a high frequency of copy number amplifications and a very low number of point mutations in HCC nodules. Copy number variations occurrence was directly correlated to the grade of malignancy in each lesion. The JNK pathway was shown to be pervasively targeted by gene amplification, and to be involved in the adenoma-to-carcinoma transition. A comparable genetic landscape has been observed in a human liver cancer with similar etiology. In conclusion, this study shows that in a model of inflammation-driven cancer an epigenetic inflammatory signature is early acquired and maintained throughout disease progression. On the contrary, genetic alterations appear only at later stages and mainly target the JNK pathway. Future dataset integration will help clarifying chronological relationship and possible mutual interplay between mutations and epigenetic changes

PPARs as new therapeutic targets for the treatment of cerebral ischemia/reperfusion injury.

Stroke is a leading cause of death and long-term disability in industrialized countries. Despite advances in understanding its pathophysiology, little progress has been made in the treatment of stroke. The currently available therapies have proven to be highly unsatisfactory (except thrombolysis) and attempts are being made to identify and characterize signaling proteins which could be exploited to design novel therapeutic modalities. The peroxisome proliferator-activated receptors (PPARs) are ligand-activated transcription factors that control lipid and glucose metabolism. PPARs regulate gene expression by binding with the retinoid X receptor (RXR) as a heterodimeric partner to specific DNA sequences, termed PPAR response elements. In addition, PPARs may modulate gene transcription also by directly interfering with other transcription factor pathways in a DNA-binding independent manner. To date, three different PPAR isoforms, designated α, β/ δ, and γ, have been identified. Recently, they have been found to play an important role for the pathogenesis of various disorders of the central nervous system and accumulating data suggest that PPARs may serve as potential targets for treating ischemic stroke. Activation of all PPAR isoforms, but especially of PPAR γ, was shown to prevent post-ischemic inflammation and neuronal damage in several in vitro and in vivo models, negatively regulating the expression of genes induced by ischemia/ reperfusion (I/R). This paper reviews the evidence and recent developments relating to the potential therapeutic effects of PPAR-agonists in the treatment of cerebral I/R injury

Remarks on hard Lefschetz conjectures on Chow groups

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.Comment: to appear in Sciences in China, Ser. A Mathematic