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I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

You mentioned that you are interested in becoming a researcher in algebraic topology. It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. The key here is to find an advisor in algebraic geometry who publishes a lot.

For example, a friend of mine who is a recent graduate in algebraic geometry tells me that there is no kunneth formula in the theory of motifs. To me, that looks like an interesting research area for an algebraic topologist right there.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.

I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers. In fact, Griffith and Harris could be viewed as a nice introduction to Hodge Theory and Complex Analytic Geometry by Claire Voisin. BTW, Voisin has by far the best explanation of what sheaves are all about that I have ever seen. I recommend reading that before reading Griffith and Harris's explanation.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

A drawback of the discussion of Chern classes omits the intuitive Fiber bundle explanation. G-H only gives the well-known method of computing them from differential geometry.

You mentioned that you are interested in becoming a researcher in algebraic topology. It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. The key here is to find an advisor in algebraic geometry who publishes a lot.

For example, a friend of mine who is a recent graduate in algebraic geometry tells me that there is no Kunneth formula in the theory of motives. To me, that looks like an interesting research area for an algebraic topologist right there.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.

I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

You mentioned that you are interested in becoming a researcher in algebraic topology. It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. The key here is to find an advisor in algebraic geometry who publishes a lot.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.

I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

You mentioned that you are interested in becoming a researcher in algebraic topology. It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. The key here is to find an advisor in algebraic geometry who publishes a lot.

For example, a friend of mine who is a recent graduate in algebraic geometry tells me that there is no kunneth formula in the theory of motifs. To me, that looks like an interesting research area for an algebraic topologist right there.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.

"Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

Deacon John Aiken, PhD in mathematical physics, 1972.

I strongly recommend "Principles of Algebraic Geometry," by Griffiths and Harris.

This book comes the closest to covering the wide range of topics in which you are interested. At least, it comes the closest of all the books of which I am aware. I have been studying all the topics you mentioned reasonably intensively for the last 50 years, so, that's a lot of books. At least 1,000.

The title is a little misleading, this book is more about differential geometry than it is about algebraic geometry. However, it does cover what one should know about differential geometry before studying algebraic geometry. Also before studying a book like Husemoller's Fiber Bundles.

If you know a little algebraic topology - like the definition of the homology and cohomology groups - and if you have a basic understanding of holomorphic (i.e., analytic) functions of a (single) complex variable, then this might be the book for you.

It would be good and natural, but not absolutely necessary, to know Differential Geometry to the level of Noel Hicks "Notes on Differential Geometry," or, equivalently, to the level of Do Carmo's two books, one on Gauss and the other on Riemannian geometry. Of course, you need the prerequisites for do Carmo's books before you are ready for Griffith and Harris.

Regarding algebraic topology, very little is required because this book explains a lot of the basic material, such as the Kunneth formula.

Regarding several complex variables, this book picks it up from the beginning, even explaining Oka's lemma. No background is needed.

On the other hand, the book is not too advanced either. K-theory is not touched on. Homotopy theory is barely mentioned. There is no p-adic analysis or finite fields. Everything is over the real or complex numbers.

There is one requirement you mentioned that you mentioned that this book does not exactly qualify for. Namely that if give light treatments. However, if the book is so excellently written that you can skip the proofs and probably get the kind of "surface" understanding that you are looking for.

Don't let the size - 800 pages - discourage you. The page count if you omit the proofs is a lot less. Also there are some topics, like "the quadratic line complex," 70 pages, that you might not be interested in.

Kahler manifolds are introduced and discussed in the first chapter, "Foundational Material."

You mentioned that you are interested in becoming a researcher in algebraic topology. It seems to me that the most fruitful field for a researcher in algebraic topology these days is algebraic geometry. For example, category theory is involved in essential ways in a.g. Also are spectral sequences and things like that. Also "resolutions." There are probably "folk theorems" lying about that would make a good dissertation. The key here is to find an advisor in algebraic geometry who publishes a lot.

Deacon John Aiken, PhD in mathematical physics, 1972, LSU.

Louis Nirenberg was my advisor at the Courant Institute, NYU where I obtained my masters.