Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involving a certain decomposition of the small diagonal in $S^3$ also proved by Beauville and the author}. We prove an analogue of such a decomposition of the small diagonal in $X^3$ for Calabi-Yau hypersurfaces $X$ in $\mathbb{P}^n$, which in turn provides strong restrictions on their Chow ring.Comment: Final version, to appear in Geometry \& Topolog

On quantization of singular varieties and applications to D-branes

We calculate the ring of differential operators on some singular affine varieties (intersecting stacks, a point on a singular curve or an orbifold). Our results support the proposed connection of the ring of differential operators with geometry of D-branes in (bosonic) string theory. In particular, the answer does know about the resolution of singularities in accordance with the string theory predictions.Comment: LaTeX2e, 17 pages, misprints correcte

Graded infinite order jet manifolds

The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds in terms of the Grassmann-graded variational bicomplex.Comment: 30 page

ADHM construction of perverse instanton sheaves

We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.Comment: 33 page

Thirty years of Chinese reform:a historical perspective

China's reform agenda is understood in the light of a thirty-year process. Interpretations of reforms differ according to the meaning of the word. Should Mao's heirs be seen as Leninist reformists, or as reformers in the historical Chinese sense, aiming at transformation in response to the West? The implications are contradictory. The reforms have been successful, yet their term has never been clearly defined. Reforms have been described alternatively as another phase in China's cyclical history, or as a transitional phase, or even as a change towards a hybrid regime with cultural Chinese features. China has had 'true reformers', aiming at fundamental change. Their legacy is still present and Chinese society is eager for economic and social debate. However, actual reform has reached a plateau. The Party's undiminished monopoly on political ideas, its legitimization through nationalist pride, its hold over the media and communications, as well as global vindication of its cautiousness, all hint of a neo-classical restoration era. However, a new wave of reforms may question again the nature of China's political system

String Bracket and Flat Connections

Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras \Psi: \H_{2\ast} (L M) \to {\o}(\Mc), where \H_{2\ast} (LM) is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and \Mc is the Maurer-Cartan moduli space of the graded differential Lie algebra \Omega^\ast (M, \adp), the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on \H_{2\ast}(LM) depending on the choice of the linear reductive Lie group $G$ in our discussion.Comment: 28 pages. This is the final versio

The Boardman-Vogt resolution of operads in monoidal model categories

We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed sigma-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction

A direct cerebello-telencephalic projection in an electrosensory mormyrid fish

After injections of the posterior part of the lateral zone of the area dorsalis telencephali (Dlp) with either horseradish peroxidase or the newly available carbocyanine dye DiI, efferent cells were labeled in the valvula cerebelli of the mormyrid fish,Gnathonemus petersii. This may be a unique connection for this group of electrosensory teleosts, since no other vertebrate has ever been reported before to have a direct cerebello-telencephalic projection

Linear Koszul Duality

In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E, a "Koszul duality" equivalence between derived categories of G_m-equivariant coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and the corresponding derived intersection of the orthogonals of F_1 and F_2 inside the dual vector bundle E^*. We also propose applications to Hecke algebras.Comment: 26 pages, final version, to appear in Compositio Mathematic

China Analysis

The Chinese financial press is euro-pessimist The wave of suicides among Foxconn workers and the vacuity of Chinese trade unionis