# Category Archives: Cohomology

## Prym Varieties

Let be an unramified double cover, where is geneus . Then has genus by the Riemann-Hurwitz formula. Now, encodes lots of information about the geometry of , especially with the additional data of the theta divisor. It turns out that … Continue reading

## The Stanley-Reisner Ring

Today, we’re going to do something completely different, but which most of my peers seem not to have seen, but is a very cool application of algebraic geometry.

## Monodromy Representations

A departure from directly working with varieties, we’re going to do something that’s strictly topological (at first glance) but which really has deep and important connections with Hodge theory. We’re going to talk about monodromy and monodromy representations. Let be … Continue reading

## The Hodge Theorem

Previously, we talked a bit about the category of Hodge structures, and did some basic constructions.  However, I’d claimed that this was algebraic geometry (at least, in the categories on the post) so today, we’ll talk about a LOT of … Continue reading

Posted in Algebraic Geometry, Cohomology, Hodge Theory | 2 Comments

## Hodge Structures

Back to blogging for a bit, though likely infrequently.  Doing a new series that might count as AG from the beginning, so I’ll put it up there once I’ve got a couple done.  We’re going to start doing some Hodge … Continue reading

## The Grothendieck-Riemann-Roch Theorem, a proof-sketch

By this time I’m sure everyone whose curiousity was piqued by the statement of the Grothendieck-Riemann-Roch theorem has read it themselves. Nevertheless, in case you haven’t, I will proceed to outline the steps of the surprisingly “easy” proof.  It is … Continue reading

Posted in Algebraic Geometry, Big Theorems, Cohomology | 5 Comments

## Applications of the Schubert Calculus

Ok, this is going to be my last post in enumerative geometry for a while, as I’m kind of drifting away from the subject.  However, this one will be fun.  We’ve already established the structure of the cohomology ring for … Continue reading

## Pieri and Giambelli Formulas

It’s been a few weeks, but now I’m back and today we’ll talk about the multiplication in the cohomology ring of Grassmannians.  Though we won’t talk about the Littlewood-Richardson rule in its full glory, we will howver discuss the special … Continue reading

## The Grothendieck-Riemann-Roch Theorem, Stated

Suppose you have a proper map between smooth (quasi) projective varieties.  Then suppose you have a coherent sheaf on .  After viewing that sheaf as an element of the Grothendieck Group of coherent sheaves on , there are two things … Continue reading

## Schubert Classes and Cellular Cohomology

So, as of the last post in the series, we defined Schubert cells.  We’re going to use them to discuss the Cohomology of the Grassmannian, and to write down an explicit basis.  With an eye looking forward, next time, we’ll … Continue reading