Braid groups and Kleinian singularities

We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.Comment: Section 4 from versions 1 and 2 has been deleted due to an error in the proof of Theorem 2. Renamed the paper and rewrote the other sections to reflect this chang

A 'Darboux theorem' for derived schemes with shifted symplectic structure

We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the symplectic form $\omega'$ is standard, and the differential in $A$ is given by Poisson bracket with a Hamiltonian function $H$ in $A$ of degree $k+1$. When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(X,\omega)$ is Zariski locally equivalent to the derived critical locus Crit$(H)$ of a regular function $H:U\to{\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the underlying classical scheme of $X$ has the structure of an 'algebraic d-critical locus', in the sense of Joyce arXiv:1304.4508. In the sequels arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090, arXiv:1504.00690, 1506.04024 we will discuss applications of these results to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining 'Fukaya categories' of Lagrangians in algebraic symplectic manifolds using perverse sheaves, and we will extend the results of this paper and arXiv:1211.3259, arXiv:1305.6428 from (derived) schemes to (derived) Artin stacks, and to give local descriptions of Lagrangians in $k$-shifted symplectic derived schemes. Bouaziz and Grojnowski arXiv:1309.2197 independently prove a similar 'Darboux Theorem'.Comment: 54 pages. (v4) Final version to appear in Journal of the AM

A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,\omega)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $\varphi:U\to X$ with $U$ an affine derived scheme, such that $(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If $(X,\omega)$ is a $-1$-shifted symplectic derived Artin stack and $X'$ the underlying classical Artin stack, then $X'$ extends naturally to a 'd-critical stack' $(X',s)$ in the sense of arXiv:1304.4508. (c) If $(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf $P^\bullet_{X,s}$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and $t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of vanishing cycles $PV_{U,f}^\bullet$ of $f$. (d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive $MF_{X,s}$ in a ring of motives $\bar{\mathcal M}^{st,\hat\mu}_X$ on $X$, such that whenever $T$ is a finite type scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and ${\mathbb L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing cycle $MF^{mot,\phi}_{U,f}$ of $f$ in $\bar{\mathcal M}^{st,\hat\mu}_T$. Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-foldsComment: (v2) 61 pages. Minor corrections, foundational material on perverse sheaves shortene

Event Studies in Securities Litigation: Low Power, Confounding Effects, and Bias

An event study is a statistical method for determining whether some event—such as the announcement of earnings or the announcement of a proposed merger—is associated with a statistically significant change in the price of a company’s stock. The main inputs to an event study are historical stock returns for the companies under study, benchmark returns like the return to the broader stock market, and standard statistical tests like t-tests that are used to test for statistical significance. In securities litigation and regulation, event studies are used primarily to detect the impact of disclosures of alleged fraud on the price of a single traded security. But are event studies in securities litigation reliable? What is interesting about the use of event studies in securities litigation is that the methodology litigants use in court differs from the methodology that economists apply in their research. With few exceptions, securities litigation event studies are single-firm event studies, while almost all academic research event studies are multi-firm event studies. Multi-firm event studies are generally accepted in financial economics research, and peer-reviewed journals contain them by the hundreds. By contrast, single-firm event studies—the mainstay of modern securities fraud litigation—are almost nonexistent in peer-reviewed journals. Importing a methodology that economists developed for use with multiple firms into a single-firm context creates three substantial difficulties. First, single-firm event studies suffer from a severe signal-to-noise problem in that they lack statistical power to detect price impacts unless the price impacts are quite large. Inattention to statistical power lowers the deterrent effect of the securities laws by giving a “free pass” to some economically meaningful price impacts and may encourage more small- and mid-scale fraud than is socially optimal given the costs of litigation. Second, single-firm event studies do not average away confounding effects. While this problem is well known, some courts have unrealistic expectations of litigants’ ability to quantitatively decompose observed price impacts into those caused by alleged fraud and those unrelated to alleged fraud. Third, low statistical power and confounding effects combine to generate sizeable upward bias in detected price impacts and therefore in damages. To improve the accuracy of adjudication in securities litigation, we suggest that litigants report the statistical power of their event studies, that courts allow litigants flexibility to deal with the problem of confounding effects, and that courts and litigants consider the possibility of upward bias in the detection of price impacts and the estimation of damages

Formulation of Policy for Cyber Crime in Criminal Law Revision Concept of Bill Book of Criminal Law (A New Penal Code)

Development of national legal systems is aimed to establish the public welfare and the protection of the public. Many attempts has been carried out to renew material criminal law and those efforts results in the formulation of the concept of the draft Law Book of the Law of Criminal Law in the form of concept criminal code draft. The basic ideas in drafting rules and regulation based on the values inside the idology of Pancasila are balance among various norm and rules in society. The design concept of the New Criminal Code Act is anticipatory and proactive to formulate provisions on Crime in Cyberspace and Crime on Information and Electronic Transactions. Several issues compiled in this paper are whether the policy in formulation of cyber crime is embodied in the provisions of the current legislation and what the policies formulation of cyber crime is in the concept of the bill book of law - criminal law recently?