Yuki

yuki

“I Gave Up on Getting Developers to Act Against Their Instincts.” Yukihiro Matsumoto, Ruby Creator

Introduction We’re thrilled that our good friend Yukihiro Matsumoto, creator of the Ruby programming language, has been able to join us at RubyRussia 2019 as a speaker for the second time, having previously spoken three years ago at RubyRussia 2016. In the time that we’ve been holding the conference — now more than ten years — Ruby has undergone a great deal of evolution, and Evrone has grown and developed alongside it. Grigory Petrov, Developer Relations at Evrone, sat down with Mr Matsumoto to hear from him first-hand about being a star, the philosophy behind Ruby’s design and evolution, and a little about Japanese life and culture. The Interview This is your second visit to Russia, the first being in 2016, and since then, the number of people attending RubyRussia has doubled. Thank you so much for your talk, I really enjoyed it, and I now know the answers to a few of the questions I was going to ask you! Grigory: I saw that people weren’t just asking you questions, but also asking for pictures with you too, and even asking for autographs — you’re a star! We developers certainly like the language, and what you’re doing for the community, but are people like this all over the world, or is this something specific to Russia? People ask to take photos everywhere: in every country and at every conference, people want to take pictures with me. But I think this is the first time anybody has asked me for an autograph! Yukihiro: As the creator of a programming language, you presumably get lots of questions, suggestions, ideas, and so on. What’s the one thing you’re asked most often? Grigory: The most popular question is: "I’m from the language X community; can’t you introduce a feature from the language X to Ruby?", or something like that. And my usual answer to these requests is… "no, I wouldn’t do that", because we have different language design and different language development policies. Yukihiro: We can’t just take some feature from language X and dump it into Ruby, although in some cases, we do of course borrow ideas from other languages such Python, JavaScript and Elixir. This question gets asked again and again, but getting a positive response to it is very rare. It’s great that you’re able to say no; I find that technologies that are “designed by the community” don’t often necessarily end up with the best results. There are so many different languages and technologies, so we can really look at the best practices and implement only the things that are a good fit. Grigory: We’ve recently seen a wave of adding gradual typing to dynamically-typed languages such as JavaScript, Python and PHP. What are your thoughts on type annotations and gradual typing? Why are they so popular? And what are your general ideas and plans for type checking in Ruby 3 and beyond? For example, Rust and Go are statically-typed languages. If the software built using them is growing pretty fast, you can end up maintaining huge codebases, potentially millions of lines of code, with hundreds of team members working on each, and in those cases, type checking is a very convenient way of detecting mismatches. Yukihiro: In contrast, with dynamically-typed languages, we have to write tests in order to avoid type mismatches in our code. As software grows, the volume of tests (and burden of writing them) grows with it, and that’s really helped to drive the recent popularity of static and gradual typing. At the same time, however, static type declaration is redundant, and with a language like Ruby, we want the benefit of static type checking without the redundancy of manual type declarations. As a Ruby community and as the Ruby language, we try to satisfy both needs at the simultaneously: we’re going to have the type information file separate from the Ruby program — the Ruby program itself doesn’t contain any static type information. Instead, a separate type information file, we call it "Ruby signature file", contains the type information about the library, Gems and your programs. We’re going to provide a tool, called the "type profiler", which can collect the type information about your software. After collecting the type information for the library and your software, the type profiler has all the information it needs about all of the classes and methods, so it can then detect type contradictions — type conflicts. You can even refine the type information in your signature file by hand to afford better type checking. In future Ruby, you’ll be able to check the types statically, to some extent — but to a very limited degree. You’ll have the traditional Ruby behaviour, which does type checking at runtime. With a "level one type checker" it's possible to find between 40-80% of type errors by using the type information available in the source code. A "level two type checker" can generate additional type information based on analysis of the code itself. With these tools, future versions of Ruby can provide static type checks without needing explicit type annotations in your code. During the talk at the conference, you’ve mentioned this, and it’s quite the idea. I’m a C/C++ developer, and so static typing is natural for me, and I always viewed types as a way of declaring my intentions, so as to help the language and tools find possible errors in the future. Grigory: Your idea is to create a system, where you write code itself, but it’s possible from that code to infer enough information to be able to spot errors without resorting explicit types, and I like the sound of that a great deal. It’s great that you conduct experiments, test ideas. What future do you see for Ruby? What way do you steer the language? Actually, I don’t steer the language or the community. I just provide technology and the community decides which way to go. At least, we have enough technology for as many domains as possible to make Ruby flexible and productive. For example, Ruby is mostly used to build web apps right now, but I’d like to see Ruby used in research, or AI and machine learning. We’re trying to make the technology usable in new domains. Yukihiro: We developers like to label and categorise things. Like, this is a sports car versus this is a family car. We label programming languages too: JavaScript is a "web language", C is a "system low-level language", C# is "for Windows UI native applications". How do you tend to categorise Ruby? Grigory: I would categorise Ruby as a productive programming language. Productivity is one of the biggest, primary goals of Ruby. I designed Ruby for humans, not for machines. Sometimes core contributors complain about the design of the language because it is difficult to implement it efficiently. Ruby’s design isn’t focused on performance first, but on productivity first. That means the implementers have a bigger challenge, but we’re excited about that challenge: we aim to make Ruby as productive for developers as possible, and, at the same time, as performant as possible. Yukihiro: With Python, we don't have multiline anonymous callables due to implementation issues. It’s good to hear that for Ruby, you and core developers are trying hard to make developers’ life easier, despite the technological challenges of the implementation. Grigory: If we talk about challenges… Imagine that you have an opportunity to travel back in time and give only one piece of advice to your younger self, at around the time you’ve started Ruby language development. What advice would you give yourself? Don’t take too much from other scripting languages. Your programming language will be the best general-purpose programming language. Features introduced just to make Ruby more comparable with scripting languages will be kind of an appendix in the future — mostly useless, until they go wrong — so it’s best not to focus too much on them. Yukihiro: In the past, there was a huge difference between "scripting" and "compiled" programming languages. Nowadays, the world has changed: between virtual machines, bytecode, and JIT compilers, the line between them is quite blurred. Grigory: During this evolution, you’ve implemented lots of changes and conducted lots of experiments with Ruby. Some were successful, some were not. What do you see as your greatest Ruby design success? What do you like the most? If I had to pick one thing, I would say blocks. A Ruby block is a quite unique, quite useful abstraction of a high-order function. It’s much simpler than in other languages. It’s restricted, but it’s convenient. Yukihiro: It’s a coincidence, but blocks are the things I like the most about Ruby. In my own talks and interviews, I say Ruby is about DSL, syntax sugar and blocks. Blocks are great. Grigory: In the languages like Swift, if the last argument is a function you can put braces around it for something like a Ruby block. There’s a proposal for a similar feature to be added to JavaScript. I’m very proud of that. Yukihiro: Yes, JavaScript can emulate blocks by passing a fat-arrow-function as the last argument, because (since ES6) it has that fat-arrow syntax for multi-line functions. Blocks are great. Meanwhile, are there things in Ruby’s design or evolution that you dislike? What would you call the biggest design failure that should be fixed, or is already fixed? Grigory: I have some. Global variables: they were useful for a "scripting language", but now they are more like an appendix. I also regret adding threads to the language, we should have had a better concurrency abstraction. My other design mistake is some objects that are needlessly mutable. For example, there is a possibility to change a time zone for the existing time object, instead of creating a new one. I regret that. Yukihiro: Yes, mutability is a tricky thing and can easily lead to errors in our source code. Grigory: Moving on from the technicalities of Ruby, how do you organize your work on the language? How does your working day look like? I’m a full-time Ruby developer. I spend half of my time thinking about the design of the future Ruby 3. The rest of the time I am working on another Ruby implementation, MRuby, along with some other projects. Yukihiro: For Ruby, the canonical Ruby implementation is done by the core contributors, and so I only need to make decisions such as "this method should behave like this", "this language feature should be like that", "the syntax like this", "the semantics are these”, and so on. I just make decisions and the other developers implement them. So, I’m half designer and half programmer. I spend about a third of my time on MRuby. I’ve checked your GitHub profile and saw lots of commits including during your flight to Russia day. Recently there were a lot of talks about developers "burning out". Do you have any free time? Any hobbies? What do you do for your well-being, in order to not burn out? Grigory: Luckily, I spend all the time on open-source software. I have no pressure from the clients. I have no bosses. I manage myself. That’s the reason I’m so relaxed at work. I have no specific deadlines besides the release date. This freedom makes me feel relaxed. That’s the one secret. The other one is that I try to spend time not related to computers and programming at all. For example, I spend time walking with my dog, petting my cat. I do some service for my local church. I spend time with my family. That time spent on my social life helps a lot. Yukihiro: Many Ruby developers from Russia like Japan as a country and enjoy Japanese culture. They watch anime, read Manga, and some of them visit Japan. As a native of Japan and a software developer, what places and experiences can you recommend to fellow developers who visit Japan? Grigory: Japan is a very diverse country. You can visit metropolitan areas like Tokyo, where it’s kind of futuristic. We have a lot of pop culture, like Manga and anime there. At the same time, we have mountains, forests and historical places, like old shrines and temples. We have beautiful cherry blossoms and the leaves. It depends on your taste and preferences, you can enjoy many things: food, nature, technology. I recommend that you enjoy the diversity of the country! Yukihiro: Is it something in Japanese culture and history that influences Ruby? Grigory: I am not sure. In Japanese, we have sentence chaining, which is similar to method chaining in Ruby, so perhaps this was influenced by the Japanese language. Japan is a wealthy country, some of us didn’t have to work so hard to sustain our lives. That allows me to work on the open-source software, which does not make money at all. We don’t sell open-source software. We can sustain our life through day jobs or from our sponsors, which can help us, including contributors, work on Ruby and to make better technology. That’s what I can think of. Yukihiro: And the last, tricky question. People often want to know what it’s like to see the world through the eyes of another person — to know what they think, and what they feel. Are there any aspects to being the language author that aren’t really obvious from outside? Grigory: The one thing is… Despite what others feel, creating the language is not really a difficult task. Computer science major students take classes for language implementation, and almost everybody that graduates can write their own programming language. Technically, it’s not that difficult. Yukihiro: At the same time, people don’t understand the nature of designing a language. Language is kind of a scaffolding of the mind, a way of structuring your thinking. It’s same to human languages, like Russian, Japanese and English. Programming languages such as Ruby, Python, JavaScript, and so on, help to develop the mind, to allow you to turn ideas into something tangible and useful. That’s the primary purpose of a programming language. I think a programming language should have a philosophy of helping our thinking, and so Ruby’s focus is on productivity and the joy of programming. Other programming languages, for example, focus instead on simplicity, performance, or something like that. Each programming language has a different philosophy and design. If you feel comfortable with Ruby’s philosophy, that means Ruby is your language. Thank you! That was Grigory Petrov and Yukihiro Matsumoto at RubyRussia 2019! Arigato! Grigory: Arigato! Yukihiro: Previously published at https://evrone.com/yukihiro-matsumoto-interview

Interview with Yukihiro Matsumoto: Ruby is Designed for Humans, not Machines

Hi! If you read my previous article about why I chose Ruby, you know why I didn’t follow JS even though it was still deep in my memory after finishing the Calculator project and the whole Foundations from The Odin Project. Introduction to Ruby with The Odin Project I have to tell you that I couldn’t wait for that moment and was super excited to learn Ruby and to really see how Yukihiro “Matz” Matsumoto managed to make it so programmer-friendly, as many say. Getting Started: Ruby Installation and Documentation Firstly I had to install everything and it wasn’t difficult with TOP guide. After a short while, everything was set and ready to go. Firstly, I was introduced to Ruby documentation, which looked a little bit different from JS. (What would I expect, though?! It’s a different language! :D) It took me some time to get used to that. Especially when it comes to all methods and classes, at that point, I could tell the OOP lesson was approaching because it seemed different from what I was used to. Once you get the hang of everything, it becomes really easy to find the information you’re looking for. The next step was to learn what IRB is. It is a Ruby interactive model that lets you type your program in the console and run it without creating any files. It is a very useful thing that Rubyists use all the time, and I have liked it since the beginning as I didn’t need to open a browser to check my code every single time like I was doing with JS. Tackling the Basics: Data Types and Initial Challenges After I was sorted with all those, I was ready to tackle basic data types, and after a couple of Pomodoro sessions, my hype began to slow down. Why? It felt super boring! I felt as if I was doing JS from the beginning. I thought I’d done this before, so why waste time doing the same thing again? I understand that I am learning a new language, and I need to start from the basics, but c’moooon! Overcoming Boredom and Staying Motivated I’ll admit I was a LITTLE disappointed at first, but I had no choice but to stick to the plan and carry on. I knew it would get more exciting at some point, but I was only afraid not to lose motivation. Why would I? It took me already so long to get to this point that there is no chance I am stopping or changing my mind. I didn’t want to fight with my thoughts any longer at this point and just decided to put in more hours and focus harder as procrastination started to crippling in, especially between my Pomodoro sessions. The next lessons were about conditionals, loops, arrays, hashes, methods, and debugging. I cannot say it was the most exciting time, and at this time, I missed making some projects. I felt as if I was forgetting how to write code while learning how to write code. Strange, isn’t it? I hope that you know what I mean. Maybe there are more of you who have had the same struggles. I then approached Basic Ruby Projects, and this news made me happy. I was finally able to give it a try and solve some bigger exercises like I used with Java Script some time ago. Well, I can’t remember how long it took me to finish them, but it wasn’t too long before I got back to TOP lessons. From Basics to Object-Oriented Programming (OOP) The next one was object-oriented programming, which I couldn’t wait for. At that moment, I knew what it was going to be all about, more or less, and I was eager to try it. Moreover, I was really curious if it’s so difficult to grasp, as people sometimes say. Since then, I planned to push through the curriculum faster during the weekends, but something was always stopping me, like my car broke, and I had to buy another one as it wasn’t worth repairing my old one. Shortly after, my family wanted to visit me, and we were planning to go to Scotland for a week. I knew already that the time I wanted to spend on learning was limited that month. When we were there, I did my best to learn Ruby for 1 hour a day at least. After we came back and I dropped my family at the airport, my friend from Birmingham came to me for the weekend as he needed me to give him a hand fixing garden furniture he produced and sold to clients not far away from where I live. On top of that, my newly bought car decided to start leaking, and to save up money, I decided to fix it myself over another weekend. Guess what… By this point, I had already lost three weeks of productivity. I was a little bit mad and scared that I wouldn’t be able to get my routine going again, but fortunately, it seemed to start picking up nicely after I started again. I continued with OOP, which didn’t seem difficult. I could see the clear, organized approach, and I really liked it! I knew, however, that once I got to 1st bigger project using this, I would probably be stuck and have to return to these lessons. Well, that’s learning, isn’t it? Ruby vs. JavaScript: My Favorite Ruby Features Apart from time management, I knew Ruby was a great choice, and even though I still had JavaScript syntax in my brain, I could see how rewarding Ruby is. It was so easy to see how things are done in Ruby, and I can promise you that if you decide to learn this, you will have plenty of AHA and ‘’why couldn’t it be implemented to JS?!” moments. Look at these examples I liked the most so far: Range Operations: In Ruby, it’s super simple! (1..5).to_a And now, look at JavaScript: let range = Array.from({length: 5}, (v, k) => k + 1); Symbol vs string usage in hashes Ruby again makes it easier to read, especially in large objects: user = { name: "Alice", age: 25 } JavaScript: let user = { "name": "Alice", "age": 25 }; Another example is a case statement. I loved it since I found out about this for the first time. I finally didn’t need to use console.log, which was quite annoying. case language
when "Ruby"
  "Great choice!"
when "Python"
  "Not bad!"
else
  "Consider Ruby!"
end Javascript switch (language) {
  case "Ruby":
    console.log("Great choice!");
    break;
  case "Python":
    console.log("Not bad!");
    break;
  default:
    console.log("Consider Ruby!");
} Tips for Staying Motivated on Your Coding Journey It took more work to get to this point of The Odin Project. I would even say that the whole Foundation was easier than the beginning of Ruby. However, after these many lessons, my willingness to learn more has returned, and I want to push further. I also stopped procrastinating so much. If you are in a similar place with your learning schedule and feeling tired and thinking about doing something else, I encourage you not to give up! Really! Maybe it sounds blatant, and I might sound like a cheap coach or something, but it works. At the moment, I am working on a Mastermind project, and it is fun. Tic-Tac-Toe helped me to refresh all my knowledge about OOP, and it seems like Mastermind feels much more fun as, at this point, I know how to solve many problems. Keep going. Everything will click at some point! I’ll keep you updated after I’m done with another challenge. Hi! If you read my previous article about why I chose Ruby, you know why I didn’t follow JS even though it was still deep in my memory after finishing the Calculator project and the whole Foundations from The Odin Project . previous article The Odin Project Introduction to Ruby with The Odin Project I have to tell you that I couldn’t wait for that moment and was super excited to learn Ruby and to really see how Yukihiro “Matz” Matsumoto managed to make it so programmer-friendly, as many say. Yukihiro “Matz” Matsumoto Getting Started: Ruby Installation and Documentation Firstly I had to install everything and it wasn’t difficult with TOP guide. After a short while, everything was set and ready to go. Firstly, I was introduced to Ruby documentation, which looked a little bit different from JS. (What would I expect, though?! It’s a different language! :D) It took me some time to get used to that. Especially when it comes to all methods and classes, at that point, I could tell the OOP lesson was approaching because it seemed different from what I was used to. Once you get the hang of everything, it becomes really easy to find the information you’re looking for. The next step was to learn what IRB is. It is a Ruby interactive model that lets you type your program in the console and run it without creating any files. It is a very useful thing that Rubyists use all the time, and I have liked it since the beginning as I didn’t need to open a browser to check my code every single time like I was doing with JS. Tackling the Basics: Data Types and Initial Challenges After I was sorted with all those, I was ready to tackle basic data types, and after a couple of Pomodoro sessions, my hype began to slow down. Why? It felt super boring! I felt as if I was doing JS from the beginning. I thought I’d done this before, so why waste time doing the same thing again? I understand that I am learning a new language, and I need to start from the basics, but c’moooon! Overcoming Boredom and Staying Motivated I’ll admit I was a LITTLE disappointed at first, but I had no choice but to stick to the plan and carry on. I knew it would get more exciting at some point, but I was only afraid not to lose motivation. Why would I? It took me already so long to get to this point that there is no chance I am stopping or changing my mind. It took me already so long to get to this point I didn’t want to fight with my thoughts any longer at this point and just decided to put in more hours and focus harder as procrastination started to crippling in, especially between my Pomodoro sessions. The next lessons were about conditionals, loops, arrays, hashes, methods, and debugging. I cannot say it was the most exciting time, and at this time, I missed making some projects. I felt as if I was forgetting how to write code while learning how to write code. Strange, isn’t it? I hope that you know what I mean. Maybe there are more of you who have had the same struggles. I then approached Basic Ruby Projects, and this news made me happy. I was finally able to give it a try and solve some bigger exercises like I used with Java Script some time ago. Well, I can’t remember how long it took me to finish them, but it wasn’t too long before I got back to TOP lessons. From Basics to Object-Oriented Programming (OOP) The next one was object-oriented programming, which I couldn’t wait for. At that moment, I knew what it was going to be all about, more or less, and I was eager to try it. Moreover, I was really curious if it’s so difficult to grasp, as people sometimes say. Since then, I planned to push through the curriculum faster during the weekends, but something was always stopping me, like my car broke, and I had to buy another one as it wasn’t worth repairing my old one. Shortly after, my family wanted to visit me, and we were planning to go to Scotland for a week. I knew already that the time I wanted to spend on learning was limited that month. When we were there, I did my best to learn Ruby for 1 hour a day at least. After we came back and I dropped my family at the airport, my friend from Birmingham came to me for the weekend as he needed me to give him a hand fixing garden furniture he produced and sold to clients not far away from where I live. On top of that, my newly bought car decided to start leaking, and to save up money, I decided to fix it myself over another weekend. Guess what… By this point, I had already lost three weeks of productivity. I was a little bit mad and scared that I wouldn’t be able to get my routine going again, but fortunately, it seemed to start picking up nicely after I started again. I continued with OOP, which didn’t seem difficult. I could see the clear, organized approach, and I really liked it! I knew, however, that once I got to 1st bigger project using this, I would probably be stuck and have to return to these lessons. Well, that’s learning, isn’t it? Ruby vs. JavaScript: My Favorite Ruby Features Apart from time management, I knew Ruby was a great choice, and even though I still had JavaScript syntax in my brain, I could see how rewarding Ruby is. It was so easy to see how things are done in Ruby, and I can promise you that if you decide to learn this, you will have plenty of AHA and ‘’why couldn’t it be implemented to JS?!” moments. Look at these examples I liked the most so far: Range Operations: In Ruby, it’s super simple! (1..5).to_a (1..5).to_a And now, look at JavaScript: let range = Array.from({length: 5}, (v, k) => k + 1); let range = Array.from({length: 5}, (v, k) => k + 1); Symbol vs string usage in hashes Ruby again makes it easier to read, especially in large objects: user = { name: "Alice", age: 25 } user = { name: "Alice", age: 25 } JavaScript: let user = { "name": "Alice", "age": 25 }; let user = { "name": "Alice", "age": 25 }; Another example is a case statement. I loved it since I found out about this for the first time. I finally didn’t need to use console.log, which was quite annoying. case language
when "Ruby"
  "Great choice!"
when "Python"
  "Not bad!"
else
  "Consider Ruby!"
end case language
when "Ruby"
  "Great choice!"
when "Python"
  "Not bad!"
else
  "Consider Ruby!"
end Javascript switch (language) {
  case "Ruby":
    console.log("Great choice!");
    break;
  case "Python":
    console.log("Not bad!");
    break;
  default:
    console.log("Consider Ruby!");
} switch (language) {
  case "Ruby":
    console.log("Great choice!");
    break;
  case "Python":
    console.log("Not bad!");
    break;
  default:
    console.log("Consider Ruby!");
} Tips for Staying Motivated on Your Coding Journey It took more work to get to this point of The Odin Project. I would even say that the whole Foundation was easier than the beginning of Ruby. However, after these many lessons, my willingness to learn more has returned, and I want to push further. I also stopped procrastinating so much. If you are in a similar place with your learning schedule and feeling tired and thinking about doing something else, I encourage you not to give up! Really! Maybe it sounds blatant, and I might sound like a cheap coach or something, but it works. At the moment, I am working on a Mastermind project, and it is fun. Tic-Tac-Toe helped me to refresh all my knowledge about OOP, and it seems like Mastermind feels much more fun as, at this point, I know how to solve many problems. Keep going. Everything will click at some point! I’ll keep you updated after I’m done with another challenge.

Falling in Love with Ruby: Getting Started with The Odin Project

NExT-GPT: Any-to-Any Multimodal LLM: Conclusion and References

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)-type graphs. The contribution of the (α, 2)-type graphs can be computed as follows. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)- type graphs . The contribution of the (α, 2)-type graphs can be computed as follows. Contribution of the type graphs This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Lagrangian Cones of Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের ল্যাগ্রাঞ্জিয়ান শঙ্কু

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Lagrangesche Kegel torischer Bündel

Un teorema del espejo para haces tóricos no divididos: conos lagrangianos de haces tóricos

Un théorème du miroir pour les faisceaux toriques non divisés : les cônes lagrangiens des faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लैग्रेंजियन शंकु

非分割トーリックバンドルのミラー定理: トーリックバンドルのラグランジアン円錐

비분할 토릭 번들에 대한 거울 정리: 토릭 번들의 라그랑주 원뿔

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Cones Lagrangianos de Pacotes Tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: лагранжевы конусы торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetlerin Lagrange Konileri

Định lý gương cho các bó Toric không phân chia: Nón Lagrange của các bó Toric

非分裂环面束的镜像定理：环面束的拉格朗日锥

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants. We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants . We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. Gromov-Witten invariant and its variants 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. Quantum Riemann-Roch theorem and twisted theory. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Genus-zero Gromov-Witten Theory

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: জেনাস-শূন্য গ্রোমভ-উইটেন তত্ত্ব

Ein Spiegelsatz für nicht-geteilte torische Bündel: Gattung-Null-Gromov-Witten-Theorie

Un teorema del espejo para haces tóricos no divididos: teoría de Gromov-Witten del género cero

Un théorème miroir pour les faisceaux toriques non divisés : théorie de Gromov-Witten du genre zéro

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: जीनस-शून्य ग्रोमोव-विटन सिद्धांत

非分割トーリック束のミラー定理: 種数ゼロのグロモフ-ウィッテン理論

비분할 토릭 번들에 대한 거울 정리: Genus-zero Gromov-Witten 이론

Um teorema do espelho para pacotes tóricos não divididos: teoria de Gromov-Witten do gênero zero

Зеркальная теорема для нерасщепляемых торических расслоений: теория Громова-Виттена с нулевым родом

Bölünmemiş Torik Demetler için Ayna Teoremi: Cins-sıfır Gromov-Witten Teorisi

Định lý gương cho các gói Toric không phân chia: Lý thuyết Gromov-Witten Genus-zero

非分裂环面束的镜像定理：属零的 Gromov-Witten 理论

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 5. Mirror theorem for a product of projective bundles In this section, we construct a twisted I-function for a product of projective bundles each coming from a vector bundle. The proof is based on the proof of the mirror theorem for a projective bundle [21, Theorem 1.1]. This section is independent of the previous section. By combining Theorem 4.2 with the mirror theorem (Theorem 5.1), we will establish the main result in the next section. Remark 5.8. For convenience, we list the rings to which the functions introduced in this section belong. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 5. Mirror theorem for a product of projective bundles In this section, we construct a twisted I -function for a product of projective bundles each coming from a vector bundle. The proof is based on the proof of the mirror theorem for a projective bundle [21, Theorem 1.1]. This section is independent of the previous section. By combining Theorem 4.2 with the mirror theorem (Theorem 5.1), we will establish the main result in the next section. I Remark 5.8. For convenience, we list the rings to which the functions introduced in this section belong. Remark 5.8. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for a Product of Projectives Bundle

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: প্রজেক্টিভ বান্ডেলের একটি পণ্যের জন্য মিরর উপপাদ্য

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Spiegelsatz für ein Produkt projektiver Bündel

Un teorema del espejo para haces tóricos no divididos: teorema del espejo para un producto de haces proyectivos

Un théorème miroir pour les bundles toriques non divisés : Théorème miroir pour un produit de bundle projectifs

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: प्रोजेक्टिव बंडल के उत्पाद के लिए दर्पण प्रमेय

非分割トーリックバンドルのミラー定理: 射影バンドルの積のミラー定理

비분할 토릭 번들에 대한 거울 정리: 투영 번들 제품에 대한 거울 정리

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Teorema do Espelho para um Produto de Pacotes Projetivos

Зеркальная теорема для нерасщепляемых торических расслоений: зеркальная теорема для произведения пучка проективов

Bölünmemiş Torik Paketler için Ayna Teoremi: Projektifler Paketinin Bir Çarpımı için Ayna Teoremi

Định lý gương cho các gói Toric không phân chia: Định lý gương cho một tích của gói xạ ảnh

非分裂环面束的镜像定理：射影束积的镜像定理

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের জন্য মিরর উপপাদ্য

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Spiegelsatz für torische Bündel

Un teorema del espejo para haces tóricos no divididos: Teorema del espejo para haces tóricos

Un théorème miroir pour les faisceaux toriques non divisés : Théorème miroir pour les faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लिए दर्पण प्रमेय

非分割トーリックバンドルのミラー定理: トーリックバンドルのミラー定理

비분할 토릭 번들에 대한 거울 정리: 토릭 번들에 대한 거울 정리

Um teorema do espelho para pacotes tóricos não divididos: Teorema do espelho para pacotes tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: Зеркальная теорема для торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetler için Ayna Teoremi

Định lý gương cho các gói Toric không phân chia: Định lý gương cho các gói Toric

非分裂环面束的镜像定理：环面束的镜像定理

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.


M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28.


K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617.


Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541.


Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482.


Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427.


Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. J. 147 (2009), no. 3, 377–438.


_________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912.


Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53.


Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458.


Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178.


William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998.


Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175.


Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112.


T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518.


Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534.


Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079.


_________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958.


_________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343.


Hiroshi Iritani, Quantum cohomology of blowups, 2023.


Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG].


Hiroshi Iritani and Fumihiko Sanda, private communication.


Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465.


Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136.


Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425.


Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340.


Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332.


Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187.


Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. J. 147 (2009), no. 3, 377–438. _________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG]. Hiroshi Iritani and Fumihiko Sanda, private communication. Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465. Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. 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Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: পরিশিষ্ট একটি এবং রেফারেন্স

Ein Spiegelsatz für nicht-geteilte torische Bündel: Anhang a und Referenzen

Un teorema del espejo para haces tóricos no divididos: Apéndice a y referencias

Un théorème du miroir pour les faisceaux toriques non divisés : annexe a et références

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: परिशिष्ट ए और संदर्भ

非分割トーリック束のミラー定理: 付録 a と参考文献

비분할 토릭 번들에 대한 거울 정리: 부록 a 및 참고 자료

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Apêndice a e Referências

Зеркальная теорема для нерасщепляемых торических расслоений: приложение а и список литературы

Bölünmemiş Torik Demetler için Ayna Teoremi: Ek a ve Referanslar

Định lý gương cho các gói Toric không phân chia: Phụ lục a và Tài liệu tham khảo

非分裂环面束的镜像定理：附录 a 和参考文献

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 3. Toric bundles In this section we introduce toric bundles. We first review toric varieties, and then define toric bundles by doing the construction of toric varieties in a relative setting. Note that they include toric bundles appearing in [5] [21]. We then investigate geometric structures of toric bundles: T-equivariant cohomology ring (3.2), effective curves (3.3), T-fixed loci and one-dimensional orbits (3.4). As explained in Section 1, mirror theorems for split toric bundles [5] and (non-split) projective bundles [21] are already known. We will prove a mirror theorem for (non-split) toric bundles (Theorem 6.1). This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 3. Toric bundles In this section we introduce toric bundles. We first review toric varieties, and then define toric bundles by doing the construction of toric varieties in a relative setting. Note that they include toric bundles appearing in [5] [21]. We then investigate geometric structures of toric bundles: T-equivariant cohomology ring (3.2), effective curves (3.3), T-fixed loci and one-dimensional orbits (3.4). As explained in Section 1, mirror theorems for split toric bundles [5] and (non-split) projective bundles [21] are already known. We will prove a mirror theorem for (non-split) toric bundles (Theorem 6.1). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেল

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Torische Bündel

Un teorema del espejo para haces tóricos no divididos: haces tóricos

Un théorème miroir pour les faisceaux toriques non divisés : les faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडल

非分割トーリックバンドルのミラー定理: トーリックバンドル

비분할 토릭 번들에 대한 거울 정리: 토릭 번들

Um teorema do espelho para pacotes tóricos não divididos: pacotes tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: Торические расслоения

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetler

Định lý gương cho các gói Toric không phân chia: Gói Toric

非分裂环面束的镜像定理：环面束

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