Symplectic singularities from the Poisson point of view

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M-theory on `toric' G_2 cones and its type II reduction

Contents
1. Introduction 2. G2 cones from ESD orbifolds. 2.1 Eight-dimensional toric hyperkahler cones 2.2 The distinguished locus 3. Toric ESD metrics and G2 metrics 4. Reduction to IIA 4.1 The reduction 4.2 Behavior of the IIA solution for ρ → 0 4.2.1 Asymptotics of U 4.2.2 Asymptotics for the metric components along ψ, θ and χ 4.2.3 Asymptotics of the coupling constant and RR one-form 4.3 Behavior on the horizontal locus 4.4 Behavior on the vertical locus 5. T-dual IIB description 5.1 The type IIB solution 5.2 Behavior of the IIB solution for ρ → 0 5.2.1 Asymptotics for the metric induced in the θ, χ, ψ directions 5.2.2 Asymptotic behavior of the modular parameter and NS-NS/RR two-forms 5.3 Behavior on the horizontal locus 5.4 Behavior on the vertical locus 6. The calibration 3-form 7. Conclusions 2 3 5 6 9 12 13 17 20 21 24 26 29 33 33 34 34 35 36 37 40 42

Galois theory of q-difference equations

[R-S-Z, Sau2]. The above moduli space is the algebraic tool to compute this part of the diﬀerence Galois group. It is possible to provide the vector bundle v (M ) on Eq , corresponding to a diﬀerence module M over K , with a connection ∇M . If M is regular singular, then ∇M is essentially determined by the absense of singularities and ‘unit circle monodromy’. More precisely, the monodromy of the connection (v (M ), ∇M ) coincides with the action of two topological generators of the universal regular singular diﬀerence Galois group ([vdP-S1, Sau1]). For irregular diﬀerence modules, ∇M will have singularities and there are various Tannakian choices for M → (v (M ), ∇M ). Explicit computations are diﬃcult, especially for the case of non integer slopes. The case of modules with integer slopes, has been studied in [R-S-Z]. This answers a question of G.D. Birkhoﬀ and follows ideas of G.D. Birkhoﬀ, P.E. Guenther, C.R. Adams (see [Bir]).

美国文学选择题2014

1. In 1837, Ralph Waldo Emerson made a speech entitled _______ at Harvard, which was hailed by Oliver Wendell Holmes as "Our intellectual Declaration of Independence."A. "Nature"B. "Self-Reliance"C. "Divinity School Address"D. "The American Scholar"2. For Melville, as well as for the reader and _______ , the narrator, Moby Dick is stilla mystery, an ultimate mystery of the universe.A. AhabB. IshmaelC. StubbD. Starbuck3. Most of the poems in Whitman's Leaves of Grass sing of the "mass" and the _______ as well.A. natureB. self-relianceC. selfD. life4. Naturalism is evolved from realism when the author's tone in writing becomes less serious and less sympathetic but more ironic and more _______ .A. rationalB. humorousC. optimisticD. pessimistic5. Dreiser's Trilogy of Desire includes three novels. They are The Financier, The Titan and _______ .A. The GeniusB. The TycoonC. The StoicD. The Giant6. The impact of Darwin's evolutionary theory on the American thought and the influence of the nineteenth-century French literature on the American men of letters gave rise to yet another school of realism: American ________ .A. local colorismB. imagismC. modernismD. naturalism7. It is on his _______ that Washington Irving's fame mainly rested.A. childhood recollectionsB. sketches about his European toursC. early poetryD. tales about America8. Which of the following works concerns most concentrated the Calvinistic view of original sin?A. The Wasteland.B. The Scarlet Letter.C. Leaves of Grass.D. As I Lay Dying9. We can perhaps summarize that Walt Whitman’s poems are characterized by all the following features except that they are _______.A. conversational and crudeB. lyrical and well-structuredC. simple and rather crudeD. free-flowing10. Who exerts the single most important influence on literary naturalism, of which Theodore Dreiser and Jack London are among the best representative writers?A. FreudB. Darwin.C. W.D. Howells.D. Emerson11. Mark Twain, one of the greatest 19th century American writers, is well known for his ____.A. international themeB. waste-land imageryC. local colorD. symbolism12. The period before the American Civil War is commonly referred to as _______.A. the Romantic PeriodB. the Realistic PeriodC. the Naturalist PeriodD. the Modern Period13. “The apparition of these faces in the crowd; / Petals on a wet, black bough.” This is the shortest poem written by（）.A. e.e. Cummings C. Ezra PoundB. T.S. Eliot D. Robert Frost14. In Henry James’ Daisy Miller, the author tries to portray the young woman as an embodiment of _______.A. the force of conventionB. the free spirit of the New WorldC. the decline of aristocracyD. the corruption of the newly rich15. "Two roads diverged in a yellow woodAnd sorry I could not travel both ..."In the above two lines of Robert Frost’s The Road Not Taken, the poet, by i mplication, was referring to _______.A. a travel experienceB. a marriage decisionC. a middle-age crisisD. one’s course of life16. The Transcendentalists believe that, first, nature is ennobling, and second, the individual is _______.A. insignificantB. vicious by natureC. divineD. forward-looking17. Which of the following is not a work of Nathaniel Hawthorne’s?A. The House of the Seven Gables.B. The Blithedale Romance.C. The Marble Falun.D. White Jacket.18. _________is often acclaimed literary spokesman of the Jazz Age.A. Carl SandburgB. Edwin Arlington RobinsonC. William FaulknerD. F. Scott Fitzgerald19. In Hawthorne’s novels and short stories, intellectuals usually appear as _______.A. commentatorsB. observersC. villainsD. saviors20. Besides sketches, tales and essays, Washington Irving also published a book on ______, which is also considered an important part of his creative writing.A. poetic theoryB. French artC. history of New YorkD. life of George Washington21. In Fitzgerald’s The Great Gatsby, there are detailed descriptions of big parties. The purpose of such descriptions is to show _______.A. emptiness of lifeB. the corruption of the upper classC. contrast of the rich and the poorD. the happy days of the Jazz Age22. In American literature, escaping from the society and returning to nature is a common subject. The following titles are all related, in one way or another, to the subject except _______.A. Mark Twain’s The Adventures of Huckleberry FinnB. Dreiser’s Sister CarrieC. Copper’s Leather-Stocking TalesD. Thoreau’s Walden23. Which of the following novels can be regarded as typically belonging to the school of literary modernism?A. The Sound and the FuryB. Uncle To m’s Cabin.C. Daisy Miller.D. The Gilded Age.24. Emily Dickinson wrote many short poems on various aspects of life. Which of the following is not a usual subject of her poetic expression?A. Religion.B. Life and death.C. Love and marriage.D. War and peace.25. Most recognizable literary movement that gave rise to the twentieth-century American literature, or we may say, the second American Renaissance, is the _______ movement.A. transcendentalB. leftistC. expatriateD. expressionistic26. As an autobiographical play, O'Neill's _______ （1956）has gained its status as a world classic and simultaneously marks the climax of his literary career and the coming of age of American drama.A. The Iceman ComethB. Long Day's Journey Into NightC. The Hairy ApeD. Desire Under the Elms27. Apart from the dislocation （错位）of time and the modern stream-of-consciousness, the other narrative techniques Faulkner used to construct his stories include _______ , symbolism and mythological and biblical allusions.A. impressionismB. expressionismC. multiple points of viewD. first person point of view28. Stylistically, Henry James' fiction is characterized by _______ .A. short, clear sentencesB. abundance of local imagesC. ordinary American speechD. highly refined language29. Robert Frost combined traditional verse forms with a plain speech of _______ farmers .A. SouthernB. WesternC. New HampshireD. New England30. Henry David Thoreau's work, ________has always been regarded as a masterpiece of New England Transcendentalism.A. WaldenB. The pioneersC. NatureD. Song of Myself31. The famous 20-years sleep in “Rip Van Winkle” helps to construct the story in such a way that we are greatly affected by Irving's ___.A. concern with the passage of timeB. expression of transient (短暂的）beautyC. satire on laziness and corruptibility of human beingsD. idea about supernatural manipulation of man's life32.Walt Whitman was a pioneering figure of American poetry. His innovation first of all lies in his use of __, poetry without a fixed beat or regular rhyme scheme.A. blank verseB. heroic coupletC. free verseD. iambic pentameter33. In Moby-Dick, the white whale symbolizes _______ for Melville, for it is complex, unfathomable, malignant, and beautiful as well.A. natureB. human societyC. whaling industryD. truth34. Hester, Dimmsdale, Chillingworth and Pearl are most likely the names of the characters in ___.A. The Scarlet LetterB. The House of the Seven GablesC. The Portrait of a LadyD. The pioneers35. With Howells, James, and Mark Twain active on the literary scene, _______ became the major trend in American literature in the seventies and eighties of the 19thcentury.A. sentimentalismB. romanticismC. realismD. naturalism36. After The adventures of Tom Sawyer, Twain gives a literary independence to Tom's buddy Huck in a book entitled ___.A. Life on the MississippiB. The Gilded AgeC. The Adventures of Huckleberry FinnD. A Connecticut Yankee in King Arthur's Court37. Generally speaking, all those writers with a naturalistic approach to human reality tend to be _____.A. transcendentalistsB. idealistsC. pessimistsD. impressionists38.In the last chapter of Sister Carrie, there is a description about Hurstwood, one of the protagonists of the novel, “Now he began leisurely to take off his clothes, but stopped first with his coat, and tucked it along the crack under the door. His vest he arranged in the same place.” Why did he do this? Because ________.A. he wanted to commit suicideB. he wanted to keep the room warmC. he didn’t want to be found by othersD. he wanted to enjoy the peace of mind39.The Romantic writers would focus on all the following issues EXCEPT the ___ in the American literary history.A .individual feelingsB. idea of survival of the fittestC. strong imaginationD. return to nature40. Chinese poetry and philosophy have exerted great influence over ____.A. Ezra PoundB. Ralph Waldo EmersonC. Robert FrostD. Emily Dickinson41. The Hemingway Code heroes（硬汉形象）are best remembered for their __.A. indestructible spiritB. pessimistic view of lifeC. war experiencesD. masculinity (男性，男子气）42. IN The Emperor Jones and The Hairy Ape, O'Neill adopted the expressionist techniques to portray the _____ of human beings in a hostile universe.A. helpless situationB. uncertaintyC. profound religious faithD. courage and perseverance43. The high tide of Romanticism in American literature occurred around .[A]1820[B]1850[C]1880[D]192044.The subj ect matter of Robert Frost’s Poems focuses on .[A] ordinary country people and scenes[B]battle scenes of ancient Greek and Roman legends[C]struggling masses and crowded urban quarters[D]fantasies and mythical happenings45.Which group of writers are among those who may be called early pioneers of American literature?[A]Mark Twain and Henry James.[B]Fenimore Cooper and Washington lrving.[C]Ernest Hemingway and William Faulkner[D]Jack London and O’Henry.46.To Theodore Dreiser, life is “so sad, so strange, so mysterious and so inexplicable.” No wonder the characters in his books are often subject to the control of the natural forces, especially those of _____and heredity.[A]fate[B]morality[C]social conventions[D]environment47.Hawthorne generally concerns himself with such issues as in his fiction.[A]the evil in man’s heart[B]the material pursuit[C]the racial conflict[D]the social inequality48._______ provides the main source of influence on American naturalism.[A]The puritan heritage[B]Howells’ ideas of realism[C]Darwin’s theory of evolution[D]The pioneer spirit of the wild west49.In Mark Twain’s The Adventures of huckleberry Finn, Huck writes a letter to inform against Jim, the escaped slave, and then he tears the letter up. This fact reveals that______ .[A]Huck has a mixed feeling of love and hate[B]there is a conflict between society and conscience in Huck[C]Huck is always an indecisive person[D]Huck has very little education50.Which terms can best describe the modernists’ concern of the human situation in their fiction?[A]Fragmentation （崩溃）and alienation.[B]Courage and honor.[C]Tradition and faith.[D]Poverty and desperation.51.Whitman’s poems are characterized by all the following features except .[A]a strict poetic form[B]a simple and conversational language[C]a free and natural rhythmic pattern[D]an easy flow of feelings52.All his novels reveal that, as time went on, Mark Twain became increasingly ____.[A]prolific (多产的）[B]artistic.[C]optimistic[D]pessimistic53.Which of the following is NOT a typical feature of Henry James’s writing style?[A] exquisite and elaborate language[B]minute and detailed descriptions[C]lengthy psychological analyses[D]American colloquialism54.In the beginning paragraph of Chapter 3, The Great Gatsby, Fitzgerald describes a big party by saying that “men and girls came and went like moths.” The author most likely indicates that______ .[A]there was a crowd of party-goers[B]such life does not have real meaning[C]these people were light-hearted[D]these were crazy and ignorant characters55.Which one of the following statements is NOT true of William Faulkner?[A]He is master of stream-of-consciousness narrative.[B]His writing is often complex and difficult to understand.[C]He often depicts slum life in New York and Chicago.[D]He represents a new group of Southern writers.56._________is generally regarded as the forerunner of the 20th century “stream-of-consciousness” novels and the founder of psychological realism.A. Theodore DreiserB. William FaulknerC. Henry JamesD. Mark Twain57.By the end of Sister Carrie, Dreiser writes, “It was forever to the pursuit of that radiance of delight which tints the distant hilltops of the world.” Dreiser implies that_____ .[A]there is a bright future lying ahead[B]there is no end to man’s desire[C]one should always be forward-looking[D]happiness is found in the end58. At the beginning of Faulkner’s A Rose For Emily, there is a detailed description of Emily’s old house. The purpose of such description is to imply that the person living in it ______.A. is a wealth ladyB. has good tasteC. is a prisoner of the pastD. is a conservative aristocrat59. ________ is often acclaimed literary spokesman of the Jazz Age.A. Carl SandburgB. Edwin Arlington RobinsonC. William FaulknerD. F. Scott Fitzgerald60.The theme of Washington Irving’s Rip Van Winkle is（）.A. the conflict of human psycheB. the fight against racial discriminationC. the familial conflictD. the nostalgia（怀旧之情）for the unrecoverable past61.Hemingway once described Mark Twain’s novel ______ the one book from which “all modern American literature comes.”A. The Adventures of Huckleberry FinnB. The Adventures of Tom SawyerC. The Gilded AgeD. The Man That Corrupted Hadleyburg。

新视野大学英语第三版读写教程Book2课文翻译

Unit 1 Text A一堂难忘的英语课1如果我是唯一一个还在纠正小孩英语的家长，那么我儿子也许是对的。

对他而言，我是一个乏味的怪物：一个他不得不听其教诲的父亲，一个还沉湎于语法规则的人，对此我儿子似乎颇为反感。

2我觉得我是在最近偶遇我以前的一位学生时，才开始对这个问题认真起来的。

这个学生刚从欧洲旅游回来。

我满怀着诚挚期待问她：“欧洲之行如何？”3她点了三四下头，绞尽脑汁，苦苦寻找恰当的词语，然后惊呼：“真是，哇！”4没了。

所有希腊文明和罗马建筑的辉煌居然囊括于一个浓缩的、不完整的语句之中！我的学生以“哇！”来表示她的惊叹，我只能以摇头表达比之更强烈的忧虑。

5关于正确使用英语能力下降的问题，有许多不同的故事。

学生的确本应该能够区分诸如their/there/they're 之间的不同，或区别complimentary 跟complementary 之间显而易见的差异。

由于这些知识缺陷，他们承受着大部分不该承受的批评和指责，因为舆论认为他们应该学得更好。

6学生并不笨，他们只是被周围所看到和听到的语言误导了。

举例来说，杂货店的指示牌会把他们引向stationary （静止处），虽然便笺本、相册、和笔记本等真正的stationery （文具用品）并没有被钉在那儿。

朋友和亲人常宣称They've just ate。

实际上，他们应该说They've just eaten。

因此，批评学生不合乎情理。

7对这种缺乏语言功底而引起的负面指责应归咎于我们的学校。

学校应对英语熟练程度制定出更高的标准。

可相反，学校只教零星的语法，高级词汇更是少之又少。

还有就是，学校的年轻教师显然缺乏这些重要的语言结构方面的知识，因为他们过去也没接触过。

学校有责任教会年轻人进行有效的语言沟通，可他们并没把语言的基本框架——准确的语法和恰当的词汇——充分地传授给学生。

8因为语法对大多数年轻学生而言枯燥且乏味，所以我觉得讲授语法得一步一步、注重技巧地进行。

'If a tree falls in a forest and no one is around to hear

Richie Furay is a founding member of legendary rock bands, Buffalo Springfield, Poco, SHF (Souther, Hillman Furay,) and currently The Richie Furay Band. He was inducted into the Rock & Roll Hall Of Fame in 1997. He has been the senior pastor of Calvary Chapel (Broomfield/Boulder) for the past two and a half decades. CALVARY CHAPEL BROOMFIELD CELEBRATES 25 YEARSThe Genesis Story of 25 Years (So Far) Of The Ministry Of Richie Furay."If a tree falls in a forest and no one is around to hear it, does it make a sound?"The origin of the question is unknown, but the current phrasing appears to haveoriginated in the twentieth century. A 1910 physics book asks: "When a tree falls in a lonely forest, and no animal is near by to hear it, does it mak e a sound?”Some of the debates surrounding this philosophical query center around mental gymnastics such as the following deep thoughts by some obviously very “enlightened” people, resulting in some fairly quantum theories:What is the difference between what something is, and how it appears? - e.g., "…sound is the variation of pressure that propagates through matter as a wave…“ "Perhaps the most important topic the riddle offers is the division between perception of an object and how an object really is. If the tree exists outside of perception (as common sense would dictate), then it will produce sound waves. However, these sound waves will not actually sound like anything. Sound as it is mechanically understood will occur, but sound as it is understood by sensation will not occur.”This riddle illustrates John Locke's famous distinction between primary and secondary qualities. This distinction outlines which qualities are actually in an object, and which qualities are not. That is, a red thing is not really red, a sweet thing is not really sweet, a sound does not actually sound like anything, but a round object is actually round.ANDCan something exist without being perceived? - e.g. “Sound is only sound if a person hears it." The most immediate philosophical topic that the riddle introduces involves the existence of the tree (and its sound) outside of human perception. If no one is around to see, hear, touch or smell the tree, how could its existence occur? What is it to say that it exists when such an existence avoids all knowing? George Berkeley in the 18th century developed subjective idealism, a metaphysical theory to respond to these questions, coined famously as "to be is to be perceived". Today metaphysicians are split. According to substance theory, a substance is distinct from its properties. According to bundle theory ,an object is merely its sense data.Well, I‟m no metaphysician but the answer to the question, "If a tree falls in a forest and no one is around to hear it, does it make a sound?" is easy, it‟s “yes.”What follows is a true story with witnesses, giving proof of the afirmative answer, and also demonstrating that the Supernatural is Master of the metaphysical.(Tree-huggers may need parental discretion…)Somewhere years ago, a tree grew. Actually two trees grew. Maybe in the same forest. Maybe not. Wherever their origin or location when they reached maturity isn‟t important. They grew where they were planted. As trees, it was their job to do so. Each eventually “fell” or ‟was” felled” by Larry the Lumberjack for what would appear to be simply for the retail purposes of man. In reality, they were grown and harvested for a Divine purpose in the free economy of God.OK, so, a couple of trees were cut down by Larry the Lumberjack, probably causing a higher demand on Arbor Day and damaging the environment irreparably… …They found their way into the bed of a semi, where Trucker Tom who, as was his job, transported the logs to a dude at the sawmill named Sam. Sam‟s job was to hone the logs into slabs of lumber suitable for some type of architecture and to destine them for a project yet unrevealed.Distributor Dan inventories into his warehouse the lumber which had been delivered by Tom the Trucker to Sam at the sawmill, who had refined the raw trunks that had been hewn by Lumberjack Larry.Of all the trees cut and manufactured in the world that day, these two sources of building materials were shipped from Distributor Dan to Broker Bob, who was the middle-man between Distributor Dan and Wayne‟s Wood Shop, where each month Jake the guitar maker would come looking for choice cuts of wood to be used in his craft. Jake‟s job is to turn trees into guitars. He chose the byproducts of these two trees. One had produced timber whose wood was blonde and smooth, the other was darker and rich with a hint of auburn, probably rosewood.According to those theories from the metaphysicians, one was not blonde and the other was not dark and rich with a hint of auburn at all. Those properties are qualities of the wood, not the wood itself- the …substance‟ simply reflects all other color so that the eye sees o nly those colors which light does not reflect. If it weren‟t being observed, it would perhaps not even exist at all anyhow, much less be wood… …OR“ the limited hues in the color spectrums which are available to translation by thehuman optical lens receptors which dictate and define perception, are then relayed through the stored databases in the cerebral cortex which controls objectidentification …these impulses are then recognized by the available vocabularyassociations stored in the language center of the brain to be used as both informationand output….the objects appear to resonate colors varying in degree as to cause the perceived object to appear to vacillate in shade from dark to light and the organic properties indicate a botanic origin…“Or in the language of the simple-minded, what was produced was a single wood body formed by two different wood sources from two unique tree types, giving the end product a distinct vertical striping of blond and brown.looking two-toned B-B ender Telecaster!Only forty stock bodies would come from the bellyof these two forever-married tree trunks,joined together by Jake the Guitar Maker.One of the forty guitars would find itself caressed by the gifted hands of Mr. Al Perkins. Al is one of the best and well known session guys in the business and has played with everybody.No, really- everybody. Al grew where he was planted.He‟s played with, recorded for, or toured with Gram Parsons, The Flying Burrit o Brothers, Manassas, Chris Hillman, Steve Stills, Michael Nesmith, Roger McGuinn, The Eagles, The Desert Rose Band, The Nash Ramblers, Dolly Parton, TJ Klay, The Rolling Stones, Emmylou Harris…He‟s played and toured with artists as diverse as Bob Dylan, Cher, Rita Coolidge, Dan Fogelberg and Dwight Yoakam.He plays pedal steel, lap steel, dobro, banjo, electric and acoustic guitars.He‟s one of those rare artists who don‟t just play really well, but who actually speak through the instrument. It‟s Al‟s in visible self you hear, not an instrument.His musical gifting seems to have no boundaries. Al is a Christian man.He speaks in tongues through his music, and the Holy Spirit translates it into the listener‟s heart.But he had a greater gifting than any of that, even.Al was playing with Manassas in the early …70‟s and then they disbanded like a lot ofbands did, and do.Chris Hillman started gigging with Richie Furay (founding member of Buffalo Springfield and Poco.) Richie got together Chris and JD Souther (co-writer of Eagles hits, “New Kid In Town,” and “Best of My Love,” among others) to form The Souther Hillman Furay Band, or SHF to their fans.When the guys wanted to add a steel pedal player, Chris brought Al Perkins along to a rehearsal. Al came to the audition, bringing along his beautiful two-toned electric. Being a Christian man, he had consecrated it with a silver “fish” emblem, which bore the testimonial words, “Jesus Is LORD”.Now let‟s recap: Maybe 25 years or so before, God had forme d those trees in the ground and they grew and were harvested for timber by Lumberjack Larry and carried in transit by Tom Trucker to Sam at the sawmill, who processed and planed them and sent them out on an order placed by Distributor Dan, who then "net-thirtied" them to Broker Bob at the request of Woodshop Wayne, where Jake the Guitar Maker would write a check and lovingly bring the pieces to the workbench of his carpentry, and create forty custom axes from his purchase. One of the forty came into the possession of Mr. Al Perkins, who was a friend of God. And Al gave glory to Jesus by branding the guitar with his conviction of faith, “Jesus Is LORD.”Then Al took Chris up on his invitation to exhibit his skills, bringing along his beautiful, two-toned custom guitar with the fish on it, and played for Richie and the other guys in SHF, and all the cavemen agreed, “Al GOOD. Al VERY good!”As with the legendary bands Buffalo Springfield and Poco, Richie Furay was at the center of the inception of this band called Souther, Hillman, Furay and he loved what he heard in Al‟s playing.But… he didn‟t love what he heard from the little fish.Richie proceeded to step the other “founding” band personnel aside and privately agreed with them that Al‟s prowess on the fret board was undeniable.It was just too darned bad about that little fish thing standing in the way and marking and marring an otherwise remarkable instrument. If only Al weren‟t a Christian, he‟d have been a great fit for SHF…Richie knew the band members would understand and agree with him that they would keep looking for another guy, a guy just like Al, but one without a fish.That day Mr. Furay had his first lesson in “God Always Gets His Way 101”. God troubled the waters and the net was cast. Al became a member of the band, despite Richie‟s objections.Over time, an abiding friendship began to grow where it was planted, and Al began to find a way into Richie‟s heart. Patiently and with long suffering, (smile) he quietly and steadfastly ministered to and witnessed to this man whose confidence was in himself alone. Al cared for this newfound brother with the compassion of someone whoalready knew in some prophetic way that can only be God, the calamity that was about to take place in Richie‟s he art.And he made sure that Richie would know who to turn to when himself and everything else failed him.Richie finally took Al‟s 70th (or 7x70th…) invitation to confess Jesus as his Salvation, to testify the Jesus IS Lord.Saint Al Perkins and the little fish on a two-toned Telecaster guitar made from a couple of trees that fell in the forest, led Richie Furay straight to the waiting hand of the Savior.The kid who was afraid of a fish emblem celebrated the 25th year of his ministry as a pastor and a servant of Christ on May 10th , 2008.The Lord‟s objective with those two trees became clear. LOUD and clear.Therefore, the answer to the question “if a tree falls in the forest and nobody is around to hear, does it make a sound,” is clearly “YES”!And the echo can be heard by a multitude.And, the Supernatural will always be Master over the metaphysical.________________________________________________________________ Riche, along with his wife of 41 years, Nancy, have served the Lordthrough their ministry at Calvary Chapel Boulder/Broomfield. May 10th ,the church celebrated 25 years of fellowship and of Richie‟s servicetherein. Al Perkins was there with …the guitar‟, along with TJ Klay and fourteen Calvary Chapel pastors from around the country.Visit for more information aboutCalvary Chapel, its history, leadership, worship times and activities and to view a musical retrospective.Visit Richie‟s personal website and find info on his books “For What It‟s Worth” and “Pickin‟ Up The Pieces” as well as CD‟s, live shows and more. 。

BASICS OF QCD PERTURBATION THEORY

ion gives xi =
i
2(
pi ) · q = 2. s
(3)
Thus only two of the xi are independent. Let θij be the angle between the momenta of partons i and j . We can relate these angles to the momentum fractions as follows: 2p1 · p2 = (p1 + p2 )2 = (q − p3 )2 = s − 2q · p3 , 2E1 E2 (1 − cos θ12 ) = s(1 − x3 ). of partons, we obtain three relations for the angles θij : x1 x2 (1 − cos θ12 ) = 2(1 − x3 ), x2 x3 (1 − cos θ23 ) = 2(1 − x1 ), x3 x1 (1 − cos θ31 ) = 2(1 − x2 ). We learn two things immediately. First, xi < 1. (7) (6) (4) (5)
2
Electron-positron annihilation and jets
In this section, I explore the structure of the ﬁnal state in QCD. I begin with the kinematics of e+ e− → 3 partons, then examine the behavior of the cross section for e+ e− → 3 partons when two of the parton momenta become collinear or one parton momentum becomes soft. In order to illustrate better what is going on, I introduce a theoretical tool, null-plane coordinates. Using this tool, I sketch a space-time picture of the singularities that we ﬁnd in momentum space. The singularities of perturbation theory correspond to long-time physics. We see that the structure of the ﬁnal state suggested by this picture conforms well with what is actually observed. I draw a the distinction between short-time physics, for which perturbation theory is useful, and long-time physics, for which the perturbative expansion is out of control. Finally, I discuss how certain experimental measurements can probe the short-time physics while avoiding sensitivity to the long-time physics.

新视野第三版读写教程第二册 U4 课后题答案

Unit 4 – Section A●Language Focus – Words in Use1.When the police caught up with him, Mr. Foster had to (confess) that he'd broken the speed limit.2.Whenever my boss makes a decision that I don't agree with, I tell him what I really think, though it's (tempting) to make him happy by telling him his ideas are always right.3.As can be probably perceived, a manned trip to Mars may soon (commence) since scientists have achieved the manned moon mission.4. A number of countries are (coordinating) their efforts to send out food to the area worst affected by the flood.5.State governments and the colleges themselves have (granted) financial help to students with special abilities and those with financial needs.6.The committee agreed that his papers (deserved) a wider circulation because of their essential and fundamental interest to a larger audience.7.The anger and frustration (displayed) by the local people who do not understand what is happening to them will be a terrible and dangerous force.8.Americans defend the right to obtain a gun, and they have (consequently) been willing to turna blind eye to the harm that people owning guns have caused.9.As the finance ministry rejected the deal for its "lack of clarity", it (lodged) a complaint with the European Commission that this deal was against the law.10.Dogs cannot distinguish the color of traffic lights, so the master must make the decision of when it is safe to (proceed) across the road.●Word Building1.contentpressedplicated4.identify5.justify6.qualify7.illustrate8.exhibition9.extend10.interpret11.plant12.perceive1.contentedpressplicate4.identification / identified5.justification / justified6.qualification / qualified7.illustration / illustrated8.exhibit9.extension / extended10.interpretation / interpretated11.plantation / planted12.perception / perceived1.Employers have found that many young competitors have the basic and best (qualifications) for success and they are intelligent, creative, and hard-working.2.Since parents are usually the major source of social support in early life, a child's (perception) of parental love may have important effects on their life.3.The storm was the most powerful to hit Hawaii this century. It greatly destroyed sugar and coffee (plantations) .4.Scientists have developed through hard work a new material that cannot be (compressed) even under extremely high pressure.w students tend to become highly concerned with matters of proper procedure and (exhibit) an increased tendency to reason.6.He was a greedy person and never felt (contented) with what he had, so he could hardly make any close friends.7.Your fashion choices make a statement about your interests or intentions, and these statements are subject to cultural (interpretation) .8.Both sides of the negotiation should create conditions for the peaceful resolution of the issue and avoid taking actions that would (complicate) the situation further.9.The architecture students could not understand their professor's (illustration) of the new library structure until he showed them the plans.10.After an analysis of his personality, I realized that his stories were a(n) (extension) of his desires to rise to higher and higher positions.11.Car manufacturers stamp vehicle (identification) numbers at several locations on new cars to help track down the vehicles in case they are stolen.12.The defense lawyer gave many excellent examples to show that the police officer had acted in self-defense and shot the man with (justification) .●Banked ClozeTraditional dating is a self-paced, general meeting of two people. The two usually(1) (commence) with spending extra time together, getting to know one another and seeing how (2) (tempting) they could be to each other. A good example is a man meeting a woman and sensing her (3) (charm). He then extends a formal invitation for a date. With traditional dating, you get to go at a steady pace, (4) (allowing) yourself and your date to get to know each other through extensive contact. There are many (5) (components) that can be expected from traditional dates. Men, who are supposed to (6) (display) these traditional values, will open the door for the woman, stand up when she leaves the table, pay for everything, and (7) (proceed) to the next move. In the1950s, a man would usually ask a woman out several days ahead for a specific date and time. If she accepted, they would (8) (arrange) for a time to pick her up. He would then take her to a dinner and a movie.Today the rules of traditional dating are less clear. Twenty years ago, if a young lady asked a men out on a date, it was thought to be (9) (weird). Now, women are being encouraged to take the initiative and ask men out. A date may consist of a brief meeting at a café or a trip to the (10) (local) art museum. Men often pay on the first date, but the woman may offer to go Dutch. The traditional dating style has been found much less common now.●Language Focus – Expressions in Use1.She authorized her partner to carry out the daily responsibilities (on her behalf) when she was on her business trip.2.On hearing the latest news about your mother's ill health, I (took the liberty of) canceling your reservation at the Sheraton.3.There was an obvious indication that the police who have to enforce the new law were not (immune to) the general discontent.4.When she heard of her failure in the experiment, her eyes (were filled with) tears; whether it was of shame, frustration, or grief was difficult to tell.5.Rose knows that continuous letters from John, together with countless roses, are aimed at making her (fall in love with) him.6.(Expelled from) public school for drinking and smoking and then failing in show business as a singer, she joined her father's business 10 years ago.7.Since the great scandals in banking, many people in the country have (been pessimistic about) the prospects of economic recovery within a few years.8.The parents were quite happy to (go along with) our suggestion because it had taken their most important concerns into consideration.●Translation●➢英译汉Valentine's Day on February 14 is celebrated in various American and European countries. It is a holiday of love and romance usually by exchanging valentines or love tokens between lovers. There are different origins regarding the festival. One legend goes that the Romans put a priest named Saint Valentine into prison for refusing to believe in the Roman gods. On February 14, Valentine was put to death not only because he was Christian, but also because he had cured the jailer's daughter of blindness. The night before he was executed he wrote her a farewell letter signed "From your Valentine". Later, February 14 became a holiday for people to show affection for their loved ones. Today, people celebrate Valentine's Day in different ways, sending greeting cards and flowers, giving chocolate or other gifts, or joining in romantic dinners. The holiday has now become popular all over the world. In China the festival is also becoming increasingly popular with young people.美洲和欧洲各国都会庆祝2月14日的情人节。

A symplectic proof of Seiberg-Witten blow-up formula

a r X i v :m a t h /0011080v 1 [m a t h .A G ] 13 N o v 2000A symplectic proof of Seiberg-Witten blow-up formulaAn-Min Li 1(Department of Mathematics,Sichuan University 610064,Chengdu,Sichuan,PRC)Renhong Wang 2(Department of Mathematics,Dalian Technology of University 116023,Dalian,Liaoning,PRC)Guosong Zhao 3(Department of Mathematics,Sichuan University 610064,Chengdu,Sichuan,PRC)Quan Zheng 4(Department of Mathematics,Sichuan University 610064,Chengdu,Sichuan,PRC)AbstractIn this paper,we give a symplectic proof for Seiberg-Witten blow-up formula of four dimensional symplectic manifolds,especially we interpret a strange phe-nomenon that the genera of embedding J-holomorphic curves will decrease when we symplectically blow-up the four dimensional symplectic manifold.1IntroductionFor 4-manifolds,there is a well-knownSeiberg-Witten Blow-Up Formula:[FS,LL]Suppose X is an arbitrary smooth four dimensional manifold.If d −1CP2(L ±(2r +1)E )(1.1)whered =11Partially supported by a NSFC and a QiuShi grant.e-mail address:amli@ 2Partially supported by a NSFC.e-mail address:renhong@ 3Partially supported by a NSFC.e-mail address:gszhao@ 4Partially supported by a postdoctor foundation of Dalian University of Technology and a Youth Foundation of Sichuan University.whereψX A,g(pt,···,pt)is the Gromov-Witten invariant(see section2)and the number of”pt”in the bracket is equal to d.We call latter embedded Gromov-Witten invariants.One can use Taubes’theorem to translate Seiberg-Witten blow-up formula into a blow-up formula for embedded Gromov-Witten invariants.Then,there is a surprising phenomenon of decreasing ly,embedded Gromov-Witten invariant of certain genus(given by ad-junction formula)is equal to certain embedded Gromov-Witten invariant of blow-up manifold with SMALLER genus.Several years ago Y.Ruan observed this phenomenon and posted a problem to give a purely symplectic explanation of this genus-lost phenomenon[R].This is the main purpose of this paper.Let X be a clossed four dimensional symplectic manifold and A∈H2(X,Z),we consider the embedded Gromov-Witten invariantψX A,g(pt,···,pt),i.e.,the Riemann surfaces have genera g determined by2g−2=A·A−C1(X)·A.(1.5)then we haveTheorem1.1Let X be a closed four dimensional symplectic manifold and˜X be its sym-plectical blow-up.For any non-negative integer r,if(C1(X)·A+g−1)−12r(r−1)(1.7)Equation(1.7)explains the genus-lost bining with Taubes’theorem,we give a symplectic proof of Seiberg-Witten blow-up formula.The main technique is the theory of the relative Gromov-Witten invariant and its gluing for-mula developed by theﬁrst author and Yongbin Ruan[LR],and E.Ionel and T.Parker[IP]. Acknowledgements The authors feel indebted to Professor Yongbin Ruan for introducing the problem to them and for very valuable conversations.2Relative GW-invariantLet(X,ω)be a real2n-dimension compact symplectic manifold with symplectic formω,and Z be a symplectic submanifolds of X with real codimension2.LetΣg be a compact connected Riemann surface of genus g≥0.Suppose A∈H2(X,Z),K={k1,···,k v}a set of positive integers. Consider moduli space M=M X,ZA,l+v(g,K)of pseudo-holomorphic maps f:Σg→X with marked points x1,···,x l;y1,···,y v such that[f(Σg)]=A,and f is tangent to Z at y1,···,y v with order k1,···,k v.Denote x=(x1,···,x l),y=(y1,···,y v).Note that the intersection number#(A·Z) is topological invariant,and v j=1k j=#(A·Z).Moreover,since Z is a symplectic submanifold,if A can be expressed by the image of an nontrivial pseudo holomorphic map f:Σg→X,the intersection number#(Z·A)≥0.Similarly to the Gromov-Uhlenbeck compactiﬁcation for the(g,K),the space of relative stable pseudo-holomorphic maps,we compactify M by M X,ZA,l+vmaps(for details see[LR]).We have two natural maps:Ξg,l:M→Z v(2.2)(f,Σg,x,y,K)−→(f(y1),···,f(y v)).Roughly,the relative GW-invariants are deﬁned asψX,Z(α|β,K)=A,g,l+vand the relative GW-invariantψX,Zg,l(α|β;K)is deﬁned to be zero unlessl i=1degαi+vj=1degβj=Ind(D f).(2.5)Remark If we choose Z is empty,then we get the ordinary Gromov-Witten invarint.Suppose that H:X→R is a proper periodic Hamiltonian function such that the Hamiltonian vectorﬁeld X H generates a circle action.By adding a constant,we can assume that zero is regular value.Then H−1(0)is a smooth submanifold preserved by circle action.The quotient Z=H−1(0)/S1is the famous symplectic ly,Z has a natural induced symplectic structure from X,so we can regard Z as a symplectic submanifold of X with real codimension2. We cut X along H−1(0).Suppose that we obtain two disjoint components X±which have boundary H−1(0).We can collapse the S1-action on H−1(0)to obtain two closed symplectic manifoldsX+ ZX+∪ZX+,R)⊗H∗(X+∪ZX+,Z,c+A+,l+(g+,K+)which consists of tuple(Σg+,x+, y+,K+,f+)with properties:•Σg+has c+connected components;•f+:Σg+→X−,Z,c−A−,l−(g−,K−)which consists of tuple(Σg−,x−,y−,f−).Ac-cording to[LR],we can glue f+and f−to obtain a pseudo holomorphic map f:Σg→X.A little more precisely,we glue X−as above.If f+and f−have same periodic orbits at each end,i.e,they have same orders as they tangent to symplectic submanifold Z,we can glue the maps f+and f−as f+#f−after gluing the domain of Riemann surfaceΣg+andΣg−,which is the connected sum ofΣg+andΣg−.Then after pertubating map f+#f−,we can get a unique pseudoholomorphic map f :Σg →X .The following index addition foumula is useful to our paper,Lemma 2.2[LR]Ind(D f +)+Ind(D f −)=(2n −2)v +Ind D f .(2.6)We also need a well known fact about genus of connected sum of Riemann surfaces:Lemma 2.3The following equality is satisﬁed:g =g ++g −+v −1(2.7)where g is the genus of Σg ,g ±is the algebraic genus of Σg ±,v is the number of end,i.e.,thenumber of the points where we glue Σg +,Σg −.According to theorem 5.8of [LR],the GW-invariant ψX A,g,l(α|β)can be expressed by the rela-tive GW-invariant over each connected component.Precisely ,using the notations of [LR],supposethat C J,Ag,lis the set of indices:(1)The combinatorial type of (Σ±,f ±):{A ±i ,g ±i ,l ±i,(k ±1,···,k ±v )}, v i =1k ±i =#(A ·Z );(2)A map ρ:{y +1,···,y +v }→{y −1,···,y −v },where (y ±1,···,y ±v )denote the puncture points of Σg ±,satisfying:(i )The map ρis one-to-one;(ii )If we identify y +i and ρ(y +i ),thenΣg+Σg −forms a Riemann surface with algebraic genus g ;(iii )f +(y +i )=f −(ρ(y +i ))and they have same order of tangency;(iv )((Σg +,f +),(Σg −,f −),ρ)represents the homology class A .For given C ∈C J,A g,lsuppose that π±C ,σ±C are partitions of x ±,y ±,α±,β±induced by C .Then we have the following([LR]Lemma 5.4and Theorem 5.8)Lemma 2.4C J,Ag,lis a ﬁnite set,and ψXA,g,l (α)=C ∈C J,Ag,lψC (α),(2.8)whereψC (α)= KδI,J ψX −,Z,c −A −,g −,l −+v (α−|βJ ;K −)(π−C ,σ−C ),(2.9)where K =k 1···k v ;δIJ =δI 1J 1···δI v J v ,δI i J i are the Kronecker symbol;and let {β1,···,βs }be an orthonormal basis of H ∗(Z,R ),βI ={βI 1,···,βI v }⊂{β1,···,βs },βJ ={βJ 1,···,βJ v }⊂{β1,···,βs}.For convenience in application,we will rewrite Lemma3.4in following steps:Step1.We divide A into A+and A−such A=A+∪Z A−.Step2.SupposeΣg±have a i≥0end points with order i∈{1,···,#(A·Z)}such that i i·a i=#(A·Z),and g=g++g−+ i a i−1.Denote a=(a1,a2,···,).Step3.Suppose thatτ±=(π±,σ±)∈P x±×P y±record which marked points in{x±,y±}go on each componentΣg±1,···,Σg±c±,satisfying:(1).g±= c±i=1g±i−c±+1,g±i≥0,i=1,···,c±(2).f i:Σg+i−→X+,Z,c+A+,g+,l++v (α+|βI;a)(π+,σ+)·ψLet X be a closed four dimensional symplectic manifold and C⊂X be any curve in X.Recall thatη:C is a compact smooth Riemann surface andηis a holomorphic map which is one to one over smooth points of C.Deﬁne the virtual genus of C byπ(C)=1+1C)(3.2)i.e.,the real genus of C is the geometric genus of its desingularization curve.Then there is an important relationship betweenπ(C)and g(C),which reads:Lemma3.1Let C⊂X,theng(C)≤π(C)(3.3)with equality holding if and only if C is smooth in X.Proof:The lemma is essentially same as lemma in book[GH]p505since all discussions are lo-cal,we omit it.If d−12r(r+1)points of P1,···,P d.Then we have(1).X−=˜X=X#X+∪ZX±be J-holomorphic maps.Without the lose of generality,suppose that Σg+=∪c+i=1Σg+i,i.e.,Σg+has c+smooth connected components,and that J-holomorphic map f+i:Σg+i→2(m−1)(m−2),g−=g−1X+in 2(m i−1)(m i−2),then by Lemma3.1,we have g i≤12(m i−1)(m i−2)−c++1.(3.4)By the same reason,we haveg−≤g−1Put (3.4)and (3.5)into (2.7),then g ≤c +i =112m (m−1))+v −1≤ c +i =112m (m −1))+m −1=g +( c +i =112(m −1)(m −2))+(1−c +)(3.6)However since c +≥1and (m −1)(m −2)≥ c +i =1(m i −1)(m i −2)for m =c +i =1m i ,m i≥0,theabove inequality holds if and only ifc +=1g +=12m (m−1)v =m(3.7)According to Lemma 3.1,this claim shows that only smooth Riemann surfaces can contribute to the Gromov-Witten invariants.In Taubes’paper [T],he denotes these smooth Riemann surfaces by symplectic submanifolds.Claim 2m =r.Note that v =m in Claim 1implies that the J-holomorphic map f ±:Σg ±→X +,which passes through 1X +.Suppose f +are tangent to Z at m 1ﬁxed points and at m −m 1non-ﬁxed points,then by the dimension condition equation (2.5),we have1X +)(mH )+2(g +−1)−2m +2(12(m−1)(m −2)and C 1(X +,Z,rA +,g +(pt,···,pt |pt,···,pt ;1,···,1)=1.(2).ψ˜X ˜A,˜g (pt,···,pt )=ψ˜X,E,1˜A,˜g (pt,···,pt |E,···,E ;1,···,1),where E ∈H 2(E,Z )∼=Z is the generator whose Pincar´e dualis the non-ﬁxed tangent point.Proof:(1)It is obvious.(2)We perform symplectic cutting along the boundary of small neighberhood ˜Nof the exceptional divisor E ⊂˜Xsuch that the neighberhood ˜N contains no ﬁxed points which are used to deﬁne theGromov-Witten invariantψ˜X˜A,˜g(pt,···,pt)over˜X.Then CP2and˜X+ is a J-holomorphic map with properties:•[f+i(Σg+i)]=d′H i−d i E,then d′i≥d i,•f+i is tangent to H atﬁxed points Q1,···,Q s and at non-ﬁxed points Q s+1,···,Q t,then d′i≥t≥s≥0.By dimension condition equation(2.5),we haves·2+(t−s)·0=2C1(˜X+)=3H−E into(4.10),we have(2d′i−d i−1)+(t−s)+g+i=0(3.11)However since d′i≥d i,t≥s and g+i≥0,the above equality(3.11)holds if and only if d′i=d i=1= s=t and g+=0.It is obvious that such J-holomorphic map is unique.Note that d= c+i=1d i=r, we conclude that c+=r,i.e.,there exist r components inΣg+.Thus applying Lemma(2.4′),we complete Claim3.Remark3.2If d=0,applying the same technique as in the proof Claim3(2),we canﬁnd r=0,thus m=0,which implies that all non-trivial J-holomorhic curve will always stay only one side2r(r−1)(3.12)Remark3.3The technique in the proof of Claim3(2)can be also applied to calculate the Gromov-Witten invariants in Riemann surfaces[LZZ1]and in any four dimensional symplectic manifold[LZZ2],especially,the recursional formula in CP2proved by Caporaso and Harris[CH] (see also[IP]).References[CH]L.Caporaso&J.Harris,Counting plane curves of any genus,preprint,1996.[FS]R.Fintushel&R.Stern,Immersed spheres in4-manifolds and the immersed Thom conjec-ture,Turkish J.Math.19(1995),145-157.[GH]P.Griﬃths&J.Harris,Principles of algebraic geometry,A Wiley-interscience publication, 1978.[H]J.X.Hu,Gromov-Witten invariants of blow-ups along points and curves,reprint.[IP] E.Ionel&T.Parker,Gromov-Witten invariants of symplectic sums,prepint.math ag/9806013.[L] E.Lerman,Symplectic cuts,Math.Research Let.2(1985)247–258.[LL]T.J.Li&A.K.Liu,The equivalence between SW and Gr in the case where b+=1,Intern.Math.Res.Notices,7(1999),335-345.[LR]An-Min Li&Yongbin Ruan,Symplectic surgery and Gromov-Witten invariants of Calabi-Yau3-folds I,preprint.math alg-geom/9803036.[LZZ1]A.M.Li&G.S.Zhao&Q.Zheng,The number of ramiﬁed covering of Riemann surface by Riemann surface,math ag/9906053.[LZZ2]A.M.Li&G.S.Zhao&Q.Zheng,The Gromov-Witten invariants in four dimensional symplectic manifolds,reprint.[R]Y.Ruan,private communication.[T] C.H.Taubes,SW⇒Gr:From the Seierg-Witten equations to pseudo-holomorphic curves, J.Amer.Math.Soc.,9(1996),845-918.[W] E.Witten,Monoples and four-manifolds,Math.Res.Lett.,1(1994),769-796.。

800句英文绕口令(完整版)

站在突出的岩石下，我误解了他的标准立场。
43. The substantial part of the constitution about the institution of institutes is substituted.
宪法中有关设立协会的实质性部分被替换。
44. Spell smell! Very well, the well-being for human being will swell.
他再三重复说：“吃肉。”
52. Having canceled X-ray scan, the cancerous candidate on the canvas ate the idle candles in the candy can.
取消X线扫描后，帆布上的癌症候选人吃了糖果罐里的闲置蜡烛。
800句英语绕口令（完整版）
1. With my own ears I clearly heard the heart beat of the nuclear bomb.
我亲耳清楚地听到原子弹的心脏的跳动。
2. Next year the bearded bear will bear a dear baby in the rear.
利用化肥保持土地肥沃。
29. The grand commander demands thousands of sandy sandwiches.
大司令官要成千个沙色三明治。
30. I infer that he is indifferent to differentiating the offers in different conferences.
军旗旁那位有男子气概的古罗马女经理具有男子风度。

A detailed case study of the rigid limit in Special Kahler geometry using K3

a rXiv:h ep-th/981112v116J a n1998KUL-TF-98/4DFTT-1/98IFUM/604-FT hep-th/9801112A detailed case study of the rigid limit in Special K¨a hler geometry using K 3.Marco Bill´o 1,Frederik Denef 1,+,Pietro Fr`e 2,Igor Pesando 2,Walter Troost 1,∗,Antoine Van Proeyen 1,†and Daniela Zanon 31Instituut voor theoretische fysica,Katholieke Universiteit Leuven,B-3001Leuven,Belgium 2Dipartimento di Fisica Teorica dell’Universit`a ,via P.Giuria 1,I-10125Torino,Italy 3Dipartimento di Fisica dell’Universit`a di Milano and INFN,Sezione di Milano,via Celoria 16,I-20133Milano,Italy.To be published in the proceedings of the 31st International Symposium Ahrenshoop on the Theory of Elementary Particles.Buckow,Brandenburg,Germany,sept.2-6,1997.Talk presented by A.V.P.ABSTRACTThis is a r´e sum´e of an extensive investigation of some examples in which one obtains the rigid limit of N =2supergravity by means of an expan-sion around singular points in the moduli space of a Calabi-Yau 3-fold.We make extensive use of the K 3ﬁbration of the Calabi-Yau manifolds which are considered.At the end the ﬁbration parameter becomes the co-ordinate of the Riemann surface whose moduli space realises rigid N =2supersymmetry.A detailed case study of the rigid limit inSpecial K¨a hler geometry using K3.Marco Bill´o1,Frederik Denef1,Pietro Fr`e2,Igor Pesando2,Walter Troost1,Antoine Van Proeyen1and Daniela Zanon31Instituut voor theoretische fysica,Katholieke Universiteit Leuven,B-3001Leuven,Belgium2Dipartimento di Fisica Teorica dell’Universit`a,via P.Giuria1,I-10125Torino,Italy3Dipartimento di Fisica dell’Universit`a di Milano andINFN,Sezione di Milano,via Celoria16,I-20133Milano,Italy.Abstract:This is a r´e sum´e of an extensive investigation of some examples in which one obtains the rigid limit of N=2supergravity by means of an expansion around singular points in the moduli space of a Calabi-Yau3-fold.We make extensive use of the K3ﬁbration of the Calabi-Yau manifolds which are considered.At the end the ﬁbration parameter becomes the coordinate of the Riemann surface whose moduli space realises rigid N=2supersymmetry.The vector multiplet of N=2,d=4supersymmetry often occurs in the study of string dualities.This is related to the fact that N=2is the minimal supersym-metry to connect the scalars to the vectors,which in four dimensions have duality transformations between their electric and magneticﬁeld strengths.These transfor-mations are in a symplectic group,and therefore the structure of the manifold of the scalars also inherits this symplectic structure.The resulting geometry of these complex scalars is denoted as‘special K¨a hler geometry’,and exists as well for rigid as for local supersymmetry.In both cases this geometry can be realised on complex structure moduli spaces:for rigid geometry on moduli spaces of a class of Riemann surfaces(RS),for local supersymmetry(supergravity)on moduli spaces of Calabi-Yau3-folds(CY).The relevant objects which build the supersymmetric actions are ‘periods’,of1-forms over1-cycles in the case of RS,of3-forms over3-cycles in the case of CY.These forms depend on the moduli which are identiﬁed with the scalar ﬁelds of the supersymmetric theory.1Many relevant CY manifolds in string theory are K3-ﬁbrations,and we will restrict to these.That means that the manifold can be described as a(complex 2-dimensional)K3surface for which the moduli depend on the moduli of the CY but also on a complex variable,denoted asζ,which can be viewed as the third complex dimension of the CY.Thus for anyﬁxed value ofζ,the CY is a K3-manifold.As such e.g.the unique CY(3,0)-form will be represented as dζ∧Ω(2,0),where the latter is the(2,0)-form of the K3.The3-cycles of the CY manifolds on the other hand can be obtained in2diﬀerent ways.One can consider a path between two(singular) points in theζ-base space where the same K3cycle bining the2-cycle above these points leads to one type of CY3-cycles.Another one can be constructed by considering in the base space a loop around such a singular point combined again with the K32-cycle which vanishes at that point.We can often calculate the integral over the K32-cycles.Then the CY period reduces to the integral of a1-form over the1-cycle in theζplane.The latter is not yet a Riemann surface however.As already mentioned,the CY moduli space has singular points where cycles degenerate.We consider an expansion around certain singular points.The expansion is as well an expansion in moduli space as in the CY coordinates.The CY moduli zαbecome in this way a function of the expansion parameterǫand variables u i, which will become the moduli of a Riemann surface.In this way the local geometry is expanded so that a rigid special K¨a hler geometry remains.In this expansion the K3manifold reduces to an ALE manifold.By performing the2-dimensional integrals,the periods of the CY reduce to periods of an element of this class of Riemann surfaces.We have made an expansion from a supergravity model to a rigid supersymmetric one.In supergravity a rigid limit is not deﬁned a priori.In the present framework this is reﬂected in that diﬀerent singular points may give rise to diﬀerent rigid limits.In [1]a procedure was set up to reduce the CY to an ALE manifold,leading to such rigid limits.See[2]for further references.Rather than using this reduction,we computed [3]all the periods in the picture of the K3-ﬁbration.This shows explicitly how the full supergravity model approaches its rigid limit.Some cycles which do not occur in the ALE manifolds lead to periods whose contribution give in the limitǫ→0(inﬁnite Planck mass)a diverging renormalisation of the rigid K¨a hler potential.Thus these renormalisation eﬀects are included in our computation.For full references see[3]. 1Special K¨a hler geometry and CY moduli spacesFirst we summarise the relevant geometric concepts,both for the rigid and for the local case.We consider symplectic vectors V(u)(rigid),resp.v(z)(local)which are holomorphic functions of r,(resp.n)complex scalars{u i}(resp.{zα}).These are 2r-vectors for the rigid case(in correspondence with the electric and magneticﬁeld strengths),and2(n+1)-vectors in the local case(because in that case there is also the graviphoton).A symplectic inner product is deﬁned asV,W =V T Q−1W; v,w =v T q−1w,(1.1)2where Q(and q)is a real,invertible,antisymmetric matrix(we wrote Q−1in(1.1) in view of the meaning which Q will get in the moduli space realisations).The K¨a hler potential is respectively for the rigid and local manifoldK(u,¯u)=i V(u),¯V(¯u) ;K(z,¯z)=−log(−i v(z),¯v(¯z) ).(1.2) In the rigid case there is a rigid invariance V→e iθV,but in the local case there is even a symmetry with a holomorphic function:v(z)→e f(z)v(z),because this gives a K¨a hler transformation K(z,¯z)→K(z,¯z)−f(z)−¯f(¯z).Because of this local symmetry we have to introduce covariant derivatives D¯αv=∂¯αv=0and Dαv=∂αv+(∂αK)v(There exists also a more symmetrical formulation).In any case we still need one more constraint(leading to the‘almost always’existence of a prepotential),which is for rigid,resp.local supersymmetry:∂i V,∂j V =0; Dαv,Dβv =0.(1.3) There are further global requirements;for an exact formulation we refer to[4].Local special K¨a hler geometry is realised in moduli spaces of CY manifolds.Con-sider a CY manifold with h21=n.It has n complex structure moduli to be identiﬁed with the complex scalars zα.There are2(n+1)3-cycles cΛ,whose intersection ma-trix will be identiﬁed with the symplectic metric qΛΣ=cΛ∩cΣ.One identiﬁes v with the‘period’vector formed by integration of the(3,0)form over the2(n+1)cycles:v= cΛΩ(3,0);Dαv= cΛΩ(2,1)(α).(1.4) Rigid special K¨a hler geometry is realised in moduli spaces of RS.A RS of genus g has g holomorphic(1,0)forms.Now in general we need a family of Riemann surfaces with r complex moduli u i,such that one can isolate r(1,0)-forms which are the derivatives of a meromorphic1-formλup to a total derivative:γi=∂iλ+dηi;α=1,...,r≤g.(1.5) Then one should also identify2r1-cycles c A forming a complete basis for the cycles over which the integrals ofλare non-zero.We identify V= c Aλ,but it should be clear that all this is much less straightforward then in the CY moduli space.2Description of a Calabi-Yau moduli spaceWe present here the description of one of the examples which we use in[3],i.e.a CY space with n=h(12)=3.First one introduces a complex4-dimensional weighted projective space in which points are equivalence classes(x1,x2,x3,x4,x5)∼(λx1,λx2,λ2x3,λ8x4,λ12x5).The CY manifold X24[1,1,2,8,12]is a3-dimensional submanifold of this projective space,determined by a polynomial equation W=0, of degree24.The manifold which we use,X∗24[1,1,2,8,12],has global identiﬁcations2πix j≃exp(n jwhere m i∈Z Z.The most general polynomial of degree24which is invariant under these identiﬁcations depends on11parameters,i.e.moduli of the CY manifold. However,they are not independent:there are still compatible redeﬁnitions of the x variables,patible with(2.1)and the weights.This leaves at the end in this example3independent moduli.We learned the advantages of not restricting immediately to one gauge choice in this moduli space.Now the K3ﬁbration is exhibited by performing the change of variablesx0=x1x2;ζ=(x1/x2)24.(2.2) We take a partial gauge choice with one remaining scale invariance,corresponding to a rescaling of x0.Then the polynomial looks likeW=112x123+12x25−ψ0(x0x1x3x5)−12B ζ+ζ−1 −ψs.(2.4)We thus have a description as a K3manifold X∗12[1,1,4,6],with a projective moduli space{B′,ψ0,ψ1}.Here B′is a function of moduli B andψs of the CY manifold and contains the dependence on the base of the K3ﬁbrationζ.The manifolds are singular when simultaneously W=0and dW=0.For the K3this happens fora)B′=(ψ1+ψ60)2;b)B′=ψ21;c)B′=0.(2.5) The singularities then occur for a speciﬁc point on K3.The cases a)and b)are A1-type singularities.They coincide ifψ60=0,in which case the singularity becomes of type A2,and ifψ60=−2ψ1,in which case the singularity becomes of type A1×A1. We will concentrate here on theﬁrst possibility:the A2singularity.For the CY to be singular,we should also have that the derivative of W with respect toζis zero,which leads to∂B′out that working in the enlarged moduli space(where gauges are not yetﬁxed) simpliﬁes the derivation of these equations,using toric geometry in disguise without introducing all the formalism,and avoiding its higher order diﬀerential equations. The independent solutions give a basis for the periods.The periods are functions of the moduli appearing in the polynomial,which have branch points in singular points, and we have to choose the position of the cuts in this moduli space.We choose a basis of solutions in one sheet of this moduli space.Continuing the periods around such singular points,we cross the cuts,and arrive to the same values of the moduli. Reexpressing the analytically continued periods in the previously chosen basis gives rise to the monodromy matrices.A generic basis of solutions to the diﬀerential equations does,however,not correspond to integrals over integer cycles.Therefore we need also supplementary methods.In some examples we can integrate over one cycle and analytically continue.We start by integrating the(2,0)form over a cycle which is known in the neighbourhood of the‘large complex structure’singular point.Then this period is analytically continued to other regions.By following its analytic continuation we also obtain the other periods.Because we start here from an integral cycle,we obtain the monodromy matrices in an integral basis.In the example described above,the strategy which we plan to use for CY,can already be used for the K3periods themselves.Indeed in this case the K3itself is a torusﬁbration.The forms and cycles can be decomposed in forms and cycles on the torus,ﬁbred over a I P1.It has the advantage that we start from the torus,where we know already a basis of cycles and its intersection matrix.The result is that weﬁnd expressions corresponding to4K3-cycles of which one vanishes at singularity a),called vα,one at b),called vβ,and two vanish at c),which we will call tαand tβ.The points of singularity of the K3manifold each occur at two points in theζ-plane,one inside the circle|ζ|=1,and one outside.We will take the cuts from the former toζ=0,and from the latter to∞.We can then construct4CY cycles by taking the paths inζbetween the two points with the same vanishing K3-cycle and combining these with the corresponding2-cycle in K3.These are called V vα,V vβ,V tαand V tβ.On the other hand we can combine the circle|ζ|=1with the4K3cycles,obtaining the CY-cycles T vα,T vβ,T tαand T tβ.4The rigid limitConsidering then again the expansion of the moduli as in(2.6),we see that(2.4) implies that the singularities are in leading order ofǫatvα:112(u1+2u60);vβ:112u1tα,tβ:112ǫψs.(4.1) The former thus keep their position,while the latter move toζ=0andζ=∞whenǫ→0.The cycles V tαand V tβthus become inﬁnitely stretched,and the5corresponding periods will get a logǫdependence.The dependence onǫis obtained from studying theǫ-monodromies.By a complex basis transformation one can isolate the diﬀerent types of smallǫbehaviour of the periods,and one rewrites the period vector v asv=v0(ǫ)+ǫ1/3v1(u)+v2(ǫ).(4.2) The relevant term will be the v1term,which has only4non-zero components,cor-responding to the cycles V vα,V vβ,T vαand T vβ,i.e.related to the singularities whichremain atﬁniteζin the limit.v0is independent of the moduli.It contains2non-zero components,one of which starts with a constant,and the other one has a logarithmic dependence alluded to above.Finally v2(ǫ)has as lowest order terms withǫ2/3and ǫ2/3logǫ.These appear in the two remaining components of v.The intersection matrix is in this basis(complex antihermitian,not an integral basis)block diagonal in the mentioned4+2+2components.For the K¨a hler potential this leads to K=−log(−i<v,¯v>)=−log −i<v0(ǫ),¯v0(ǫ)>−i|ǫ|2/3<v1(u),¯v1(u)>+R(ǫ,u,¯u) (4.3)≈−log(−i<v0(ǫ),¯v0(ǫ)>)−|ǫ|2/3。

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we prove the structure theorems for holonomic Poisson schemes. Then we show that is a scheme is in addition locally exact, then it admits, again locally, a Gm -action. Acknowledgements. In the recent years, I have beneﬁtted from many opportunities to discuss symplectic singularities with colleages; the discussion were very fruitful and often led to better understanding on my part. I would like to thank, in particular, R. Bezrukavnikov, H. Esnault, P. Etingof, B. Fu, V. Ginzburg, D. Huybrechts, D. Kazhdan, A. Kuznetsov, M. Lehn, Y. Namikawa, M. Verbitsky, E. Viehweg, J. Wierzba – and I would like to apologize to those I forgot to mention here. The results were presented at an Oberwolfach meeting on singularities in 2003. I am grateful to the organizers, in particular to D. van Straten, for inviting me and giving me an opportunity to beneﬁt from the beautiful atmosphere of this great institution. I am also grateful to A. Bondal, B. Fu, V. Ginzburg and Y. Namikawa who have read the ﬁrst version of this paper, suggested several improvements and found some gaps in the proofs. Finally, I would like to thank the referee for a thoughtful report and important suggestions and corrections.
arXiv:math/0310186v4 [math.AG] 9 Mar 2006
Symplectic singularities from the Poisson point of view
D. Kaledin∗
Contents
1 Generalities on Poisson schemes. 2 Symplectic singularities. 2.1 Statements. . . . . . . . 2.2 Geometry of resolutions. 2.3 Proofs. . . . . . . . . . . 2.4 Symplectic resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 6 6 9 12 15 18 24
3 Stratiﬁcation and product decomposition. 4 Group actions.
Introduction
In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C ∞ symplectic manifold M . The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorphic or algebro-geometric setting, one expects the Poisson bracket to be even more useful because of the following observation: the bracket is a purely algebraic structure, and it generalizes immediately to singular algebraic varieties and complex-analytic spaces. The appropriate notion of singularities for symplectic algebraic varieties has been introduced recently by A. Beauville [B] and studieted by CRDF grant RM1-2354-MO02.
1
Namikawa [N1], [N2]. However, the theory of singular symplectic algebraic varieties is in its starting stages; in particular, to the best of our knowledge, the Poisson methods have not been used yet. This is the goal of the present paper. Our results are twofold. Firstly, we prove a simple but useful structure theorem about symplectic varieties (Theorem 2.3) which says, roughly, that any symplectic variety admits a canonical stratiﬁcation with a ﬁnite number of symplectic strata (in the Poisson language, a symplectic variety considered as a Poisson space has a ﬁnite number of symplectic leaves). In addition, we show that, locally near a stratum, the variety in question admits a nice decomposition into the product of the stratum itself and a transversal slice. Secondly, we study natural group actions on a symplectic variety and we prove that, again locally, a symplectic variety always admits a non-trivial action of the one-dimensional torus Gm (Theorem 2.4). This is a rather strong restriction on the type of singularities a symplectic variety might have. Unfortunately, the paper is much more eclectic than we would like. Moreover, one of the two main results is seriously ﬂawed: we were not able to show that the Gm -action provided by Theorem 2.4 has positive weights. However, all the results has been known to the author for a couple of years now, and it seems that any improvement would require substantially new methods. Thus we have decided to publish the statements “as is”. In addition, we separately consider a special (and relatively rare) situation when a symplectic variety admits a crepant resolution of singularities. We prove that the geometry of such a resolution is very restricted: it is always semismall, and the Hodge structure on the cohomology of its ﬁbers is pure and Hodge-Tate. Our approach, for better or for worse, is to try to use Poisson algebraic methods as much as possible, getting rid of actual geometry at an early stage. The paper is organized as follows. In the ﬁrst section we recall all the necessary deﬁnitions, both from the Poisson side of the story and from the theory of symplectic singularities. We also introduce two particular classes of Poisson schemes which we call holonomic and locally exact. In the second section we show that symplectic varieties give examples of Poisson schemes lying in both of these classes. The main technical tool here is the beautiful canonical resolution of singularities discovered in the last two decades (see, for example, [BM]). Then in Subsection 2.4, we study the geometry of crepant resolutions. The remainder of the paper is purely algebraic. First, 2