Morse theory on singular spaces

tom-kitten-bursting-his-pair-of-pantsMorse theory is a powerful tool in topology, relating the global properties of a smooth manifold X to the critical points of a smooth function on X. In this note I want to consider the possibility of Morse theory on singular spaces. Some of this dream can be made to work in algebraic geometry, where it helps to analyze the Hilbert scheme of points in new cases.

This is related to my joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson on “The Hilbert scheme of infinite affine space.” The connection with Morse theory appears in my paper Torus actions, Morse homology, and the Hilbert scheme of points on affine space.

Morse theory studies a smooth function f on a closed Riemannian manifold X using the gradient flow associated to f. That is, from any point on X, move in the direction in which f is (say) decreasing fastest. In the limit, every point of X is attracted to one of the critical points of f.

From the modern point of view known as “Morse homology,” a central part of the theory is to compactify the space of orbits of the gradient flow. The key point is that any limit of orbits of the gradient flow is a broken trajectory, a chain of orbits that connect a sequence of critical points with decreasing values of f.

morse

This situation has an analog in algebraic geometry. Consider a projective variety X with an action of the multiplicative group T = Gm. (Over the complex numbers, this group can also be called C*.) Then the orbits of T on X are analogous to the gradient flow lines in Morse theory. In particular, for every point x in X, the “downward limit” of its orbit, limt→0(tx), is a T-fixed point of X (analogous to a critical point in Morse theory). Over the complex numbers, this T-action can actually be identified with Morse theory for the “Hamiltonian function” on X.

Just as in Morse theory, I show that every limit of T-orbits on a projective variety X is a broken trajectory, a chain of orbits that connect a sequence of T-fixed points. An interesting point is that this works without assuming that X is smooth. So this is a possible model for Morse theory on singular spaces.

I give an application to the Hilbert scheme of points on affine space. Namely, let Hilbd(An) be the space of 0-dimensional subschemes of degree d in affine n-space. And let Hilbd(An,0) be the (compact) subspace of subschemes supported at the origin in An. I show that, over the complex numbers, these two spaces are homotopy equivalent. (Computing the cohomology of either space is a wide open problem, in general.) The proof uses the algebraic version of Morse theory described here, using the action of T on Hilbd(An) coming from the action on An by scaling. I hope to see more applications: torus actions are everywhere in algebraic geometry.

Image: A chubby Tom Kitten in a broken pair of pants, from The Tale of Tom Kitten, by Beatrix Potter.

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How to get started with the Hilbert scheme

a9d0c6f992d5f50fNow that we can attend seminars all over the world, beginning algebraic geometers may be encountering the Hilbert scheme everywhere. At first glance, however, the idea of the Hilbert scheme is so capacious that it can be hard to grasp.

So, in this post, I want to sketch a path that an interested reader (or student seminar) could follow in beginning to understand the Hilbert scheme. (Some of the links may require your library to have access.) The topic came to mind because of my current joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Maria Yakerson, “The Hilbert scheme of infinite affine space.”

First, the general definition: for a projective variety (or scheme) X over a field, the Hilbert scheme of X classifies all closed subschemes of X. The existence of the Hilbert scheme (as a union of projective schemes) reflects a basic feature of algebraic geometry: families of algebraic varieties are parametrized by an algebraic variety (or scheme). But then we have to find ways to analyze the Hilbert scheme in cases of interest.

To start the path of reading, there is an excellent introduction by James McKernan, a 10-page set of MIT lecture notes. It discusses a special case, the “Hilbert scheme of points”, with several examples.

The general construction of Hilbert schemes, due to Grothendieck, is outlined in several places. Perhaps the easiest to read is chapter 1 of Harris and Morrison’s book Moduli of Curves. The most technical step of the proof is not included there (roughly, the fact that there is a uniform bound for the equations of all subschemes of projective space with a given Hilbert polynomial). It might be reasonable for learners to come back to this step later. A standard reference for this step is chapter 14 of Mumford’s book Lectures on Curves on an Algebraic Surface.

Although the Hilbert scheme is hard to understand in full detail, there is a clear — computable! — description of its Zariski tangent space at any point, even where it is singular. Namely, if S is a closed subscheme of a projective scheme X over a field, then the tangent space to Hilb(X) at the point [S] is H^0(S, N_{S/X}), the space of global sections of the normal sheaf. Computing this group in examples is essential for getting to grips with the Hilbert scheme. As a start, you can look in chapter 1 of Harris-Morrison or chapter 1 of Kollár’s book Rational Curves on Algebraic Varieties.

Finally, for many applications, it is important to go one step further and understand the “obstruction space” as well as the tangent space for the Hilbert scheme. This is a great setting in which to learn deformation theory. Roughly, the obstruction space tells you the number of equations needed to define the Hilbert scheme near a given point. There are many possible introductions to deformation theory; let me recommend Sernesi’s book Deformations of Algebraic Schemes. Section 3.2 addresses the Hilbert scheme, with examples and exercises.

There is a vast literature on Hilbert schemes in particular settings, such as the Hilbert scheme of points on a surface. But I hope what I’ve said is enough for you to start exploring.

Image by @kernpanik; license CC BY-NC-SA 4.0.

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WAGON (WAGS Online), 18-19 April 2020

The Western Algebraic Geometry Symposium (WAGS) is going online and worldwide! For more details and to register, see the conference webpage.

Image lifted from @littmath.

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Late-night motives* — MoVid-20: motivic video-conference on 15 April 2020 via Zoom

*The first talk is 1 am Pacific time.

To register, please send an email to denis.nardin@ur.de with the subject “MoVid-20”.

The schedule in Central European Summer Time (aka time in Germany) is as follows.

9:45-10:00 Conference opening
10:00-11:15 Tom Bachmann
11:15-12:00 coffee break
12:00-13:15 Marc Hoyois
13:15-14:30 lunch break
14:30-15:45 Maria Yakerson
15:45-16:30 coffee break
16:30-17:45 Denis Nardin

Here are the titles and abstracts.

Tom Bachmann: Pullbacks for the Rost-Schmid complex
Let F be a “strictly homotopy invariant” Nisnevich sheaf of abelian groups on the site of smooth varieties over a perfect field k. By work of Morel and Colliot-Thélène–Hoobler–Kahn, the cohomology of F may be computed using a fairly explicit “Rost-Schmid” complex. However, given a morphism f : XY of smooth varieties, it is in general (in particular if f is not flat, e.g. a closed immersion) unclear how to compute the pullback map f *: H*(Y,F) → H*(X,F) in terms of the Rost-Schmid complex. I will explain how to compute the pullback of a cycle with support Z such that f-1(Z) has the expected dimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space X, its zeroth P1-stable homotopy sheaf is given by π3P13X)-3.

Marc Hoyois: Milnor excision for motivic spectra
It is a classical result of Weibel that homotopy invariant algebraic K-theory satisfies excision, in the sense that for any ring A and ideal I\subset A, the fiber of KH(A) → KH(A/I) depends only on I as a nonunital ring. In joint work with Elden Elmanto, Ryomei Iwasa, and Shane Kelly, we show that this is true more generally for any cohomology theory represented by a motivic spectrum.

Denis Nardin: A description of the motive of $Hilb(A^\infty)$
The Hilbert scheme of points in infinite affine space is a very complicated algebro-geometric object, whose local structure is extremely rich and hard to describe. In this talk I will show that nevertheless its motive is pure Tate and in fact it coincides with the motive of the Grassmannian. This will allow us to give a simple conceptual description of the motivic algebraic K-theory spectrum. This is joint work with Marc Hoyois, Joachim Jelisiejew, Burt Totaro and Maria Yakerson.

Maria Yakerson: Motivic generalized cohomology theories from framed perspective
All motivic generalized cohomology theories acquire unique structure of so called framed transfers. If one takes framed transfers into account, it turns out that many interesting cohomology theories can be constructed simply as suspension spectra on certain moduli stacks (and their variations). This way important cohomology theories on schemes get new geometric interpretations, and so do canonical maps between different cohomology theories. In the talk we will explain the general formalism of framed transfers and
show how it works for various cohomology theories. This is a summary of joint projects with Tom Bachmann, Elden Elmanto, Marc Hoyois, Joachim Jelisiejew, Adeel Khan, Denis Nardin and Vladimir Sosnilo.

Image: Still from The Third Man (1949, dir. Carol Reed).

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Happy Birthday to this peevish two-year-old

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March 31, 2020 · 9:55 pm

WAGS @ Pomona, POSTPONED

This kitten sneaked into Magic Door Books in Pomona last summer

WAGS at Pomona is being postponed until the Fall term.

The Spring 2020 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at Pomona College on 3–4 April 2020. You can register here. Financial assistance is available and can be requested via the registration process.

Image: Bookstore cat who works at Magic Door Bookstore IV in downtown Pomona.

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The bees and the birds at UCLA

Getting emails from central administration is often a joyless part of university life, but today’s email was interesting, informative, and even delightful.

==========
In the Spring and Summer it is common to find young birds away from their nests. Often concerned students or staff will think the birds need rescuing, but it depends. Below is a note from one of our bird researchers on campus and a helpful article from Audubon. A couple photos attached from campus showing nestlings versus fledglings. Please share this information with your departments, and let them know that if they are not sure they can call or text me: [xxx.xxx.xxxx]. Our office works with a network of wildlife rehabilitators and can help transport wildlife in need of rescue. In addition, with everything starting to bloom, you may see bees around campus. See below for instructions if you discover a hive or swarm.

Bee Information:
Sometimes bees can swarm or build hives on campus. UCLA recognizes the critical role pollinators like bees play in our food system and ecology. We work with a company to ensure that bees are live captured and relocated and not killed. See photo attached of a swarm removal from on a car. If you discover a hive on campus or swarm that needs removal please call Facilities Management Trouble call at [yyy.yyy.yyyy], they will coordinate the response.

Bird Information:
One of our bird researchers on campus notes: “Knowing when a young bird is “supposed” to be out of the nest vs. when it’s not supposed to is key when dealing with these sorts of situations. It’s important to remember that nests are unsafe places to be; it’s easier for a predator to kill four chicks that are in the same cup of sticks and hair than four chicks that are in four different parts of their parents’ territory. As a result, parents will push their chicks out of the nest before they’re fully able to fly, and take care of itself.

* If you have to chase after the chick to catch it, it’s old enough to let it’s parents take care of it outside the nest. If it’s out in the open, or in a dangerous place try herding it to the nearest shrub or other protected place. Mom and dad know where the fledgling is and will feed it discretely.
* If the bird is sitting upright and is alert, it probably has recently left the nest. Check the wings, if the wings are fully or partially feathered (as opposed to being in gray-looking sheaths), it’s old enough to be outside of the nest. If it’s out in the open, or in a dangerous place, you can move it to a place nearby with greater safety.
* Most “baby” birds you find will fit in above. In both cases, they are where they need to be. Even if it looks like they’re abandoned, they aren’t, the parents are just making sure not to lead predators to their offspring. Trying to rescue it means a lot more work and stress for you and the wildlife rehabber you take it to, when the parents will almost certainly do a better job for free.

So: when should you interfere with nature?

* Very young nestlings (ie mostly naked, no or few feathers, can’t sit up, appears helpless). It might have fallen out of the nest; if so look around to see if you can find the nest. If so, put it back, if not, follow the directions in the article cited below, and then bring the nestling to a licensed rehabber.
* The chick is obviously injured (ie broken wings or legs.) In this case, bring the chick to a licensed rehabber. If you know or suspect the chick was grabbed by a cat, bring it to a rehabber immediately! This is because cat mouths are breeding grounds for all sorts of nasty bacterial that kill birds, and any bird that has been exposed to cat teeth needs to be given antibiotics ASAP.

Some additional information can be found in this helpful article from Audubon, When You Should—and Should Not—Rescue Baby Birds: https://www.audubon.org/news/when-you-should-and-should-not-rescue-baby-birds

Thank you,
Nurit

Nurit Katz
Chief Sustainability Officer
Executive Officer of Facilities Management
UCLA

Images are (from top to bottom) junco juvenile, bee removal, junco nestlings, fledgling starlings.

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SoCalAGS @ UCSD, 29 February 2020

The next meeting of the Southern California Algebraic Geometry Seminar (SoCalAGS) will take place on Saturday, 29 February 2020, at University of California, San Diego. Speakers are:

Fabio Bernasconi (Utah)
Nathaniel Bottman (USC)
David Stapleton (UCSD)
Maria Yakerson (Regensburg)

Please register. It’s free and helps us to demonstrate appetite for this seminar.

Image via @uscg: “In 1945, Salty, the mascot at Air Station San Diego, became the first cat to take part in a rescue mission when she stowed away with her kittens on an amphibious reconnaissance plane just before it took off to rescue a pilot who had gone down at sea. #NationalCatDay #SARcat” Look carefully for the second kitten!

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What if J. Peterman wanted you to learn commutative algebra?

1983.

The crisp leaves as I waited for the Dinky. Atiyah-Macdonald light in my hand. That was the fall I became intoxicated with power as I shrank open sets to points and revealed the structure of the universe. By the holidays I’d acquired the whole set of moves. Cracking the nut of a Noetherian ring. Folding and unfolding integral extensions. Localizing to a point with a gimlet eye. Flipping from rings to spaces, spaces to rings, with the ease of Evert working the baseline.

Math 215A. Commutative Algebra. James Cameron. Not that James Cameron, but just as life-changing.

— Guest post

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WAGS @ Utah, 2–3 November 2019

The action-packed Fall 2019 meeting of the Western Algebraic Geometry Symposium (WAGS) will take place at the University of Utah in Salt Lake City.

Personal note: I’ll be attending WAGS and would be interested in chatting with anyone who has experience using MAGMA. — Burt

Speakers are:
Daniel Bragg, Berkeley
Kristin DeVleming, UCSD
Eric Ramos, Oregon
Stefan Schreieder, Munich
Isabel Vogt, Stanford
Jakub Witaszek, Michigan

Registration is free; funding is available for graduate students.

Photo shows a cat sprinting through play between Tigres UANL and Real Salt Lake City at Rio Tinto Stadium in Salt Lake City, 24 July 2019.

Photo credit: Jeff Swinger-USA TODAY Sports.

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