No, this e-mail is not good at all. The other answers have already explained why not. But you say you want a "canonical answer," so let's go through line-by-line.
In English-speaking North America and Europe, this is already a red flag. I see this often, so I assume it is the proper form of address in some English-speaking country (India?). But in North America and Europe, this is not how we address mail. Worse, the mails we do receive that begin with "respected professor" are almost never worth reading; so, we have learned to trash such mails without reading them.
The proper form of address may vary by country; "Dear Professor X" or "Hi Professor X" should be OK throughout North America and Europe. Note, you should put the actual name into this block; otherwise, I assume you are just copying-and-pasting a generic response to hundreds of people (at which point, I immediately stop reading and trash it).
I am a resident of [my Country name] and I completed my masters in
mathematics in June 2020 from [My Institute Name]. I am looking for PhD positions in Algebraic Geometry.
Here is the key point you need to internalize: most readers are just going to skim the first paragraph and then press delete. Your readers receive a ton of mail, much of it from prospective students, and do not have the time or inclination to carefully review all of them.
So, your job in this mail, and especially in this paragraph, is to convince them that:
- You are very impressive, and
- Your interests are very well-aligned
If you fail on either of these counts, they will delete without responding. In fact, a pretty high percentage will delete without responding even if they are convinced, but there's nothing you can do about that.
So, your first paragraph needs to impress them. And I mean really impress them. Something like: "I am looking for PhD positions in Algebraic Geometry. I have 3 papers in [impressive journals], have won an award from [somewhere] (the most prestigious body of mathematics in my country), have the equivalent of a 3.91 GPA, and am particularly interested in [some niche topic]." Note that you are:
- being concise,
- listing the specifics right here in the first paragraph, and
- using a language the professor can understand (e.g., converting the GPA to whatever scale is used in the professor's country, explaining that your award is from the most prestigious mathematical body in your country, etc.)
I realize that you may not have any super-impressive accomplishments like this, but you need to list whatever you do have. Put it all out there in this mail, because if you don't, you will not get another opportunity.
Since June 2020, I took a break to self study more mathematics courses
but couldnot apply anywhere in session 2021 due to personal reasons.
Nope. See above. In this e-mail, your one job is to volunteer information that will impress them. This information is not impressive, so don't volunteer it. Delete this paragraph completely.
Upon learning about your work I went through your research satement
and found it to be aligning with my interests, hence I would very much
like to work with you. Kindly find my CV attached along with this
e-mail. Can you please tell me if there is a vacancy for new PhD
applications in your working group for session 2022?
This is so generic it is completely meaningless. What part of my "research statement" aligned with your interest? Again, I assume you are just sending this to hundreds of professors. Instead, you should briefly explain why in particular you want to work for me -- which of my papers did you read? Why did you find them interesting? What related work have you done? If we worked together, what topics could we pursue?
Also, I wouldn't say "I would very much like to work with you." You don't even know me! Instead, say that you're interested in "exploring this further" or "discussing future possibilities." Yes, it may be that you are desperate and willing to take anything, but you need to conceal this; desperation is not attractive, and professors receive tons of mail from desperate people.
I am available to discuss the possiblilty further and look foreward
hearing from you.
If you are asking for a job, it goes without saying that you're available to discuss, etc. It is important that your mail be very short (just 2 or 3 short paragraphs), so don't waste any space with generic statements like "looking forward to hearing from you." Delete all this (except your name). Seriously! I realize some cultures require polite, effusive farewells, but in Western Academia, putting your name alone is perfectly fine, and it's the best option in many cases.
Finally, I would be remiss if I didn't point out that there are several misspellings in your mail. I realize English may not be your native language, but you should get your mail proofread by a native. Most mails that contain misspellings also highly correlated with mails I don't care about, and so most academics have learned to hit the delete button after the second or third error.