Let $ X $ to be a complex projective variety. Because the degree is additive in the exact sequences, it goes down to the Grothendieck group $ overline {deg}: K (X) rightarrow mathbb {Z} $. Because the degree of coherent sheaf satisfies the identity $ deg (E otimes F) = deg (E) + deg (F) $ we have the logarithmic identity $ overline {deg} ([E] cdot [F]) = overline {deg} (E) + overline {deg} (F) $. Is there any sense in which the degree can be interpreted as a logarithm?

# Tag: geometry

## dg.differential geometry – Calculates the parallel part of $ g_Y $ of a form $ (0,1) $

Let $ Y $ to be a compact (unlimited) variety of Calabi-Yau, that is to say $ c_1 (Y) = $ 0 in $ H ^ 2 (Y, mathbb {R}) $. Let $ omega $ to be a Kähler form on $ mathbb {C} ^ m times Y $ and let $ omega_P = omega _ { mathbb {C} ^ m} + omega_Y $. Suppose that $ zeta = omega – omega_P $ is an exact $ (1,1) $-form, that is to say $ zeta = d xi $ for real $ 1 $-form $ xi $ sure $ mathbb {C} ^ m times Y $. By the spectral sequence of Leray of the projection $ pi _ { mathbb {C} ^ m}: mathbb {C} ^ m times Y to mathbb {C} ^ m $there is an isomorphism $ Phi: H ^ {0,1} ( mathbb {C} ^ m times Y) to mathcal {O} ( mathbb {C} ^ m, H ^ {0,1} (Y)) $ with $$ Phi[xi^{0,1}](z) = [xi^{0,1} vert_{{ z } times Y}]. $$ Together $ Phi[xi^{0,1}]= f $ and identify $ H ^ {0,1} (Y) $ with the space of $ g_Y $-parallel $ (0.1) $Platforms.

**Q:** I want to show that $$ frac { partial f} { partial overline {z} ^ j} = 0. $$

In addition, I want to show that $ dfrac { partial f} { partial z ^ j} $ is the $ H ^ {0,1} $-class, or the $ g_Y $parallel part of $ (0.1) $-form $$ ( partial_ {z ^ j} llcorner zeta) green _ { {z } times Y}, $$ or $ llcorner $ means the inner product.

## Aggressive geometry – Singularity of Brill-Noether subvarieties of smooth curve varieties of Picard

assume $ C $ is a smooth projective curve on complex numbers. The singularities of the theta divider $ Theta $ in $ Pic ^ {g-1} (C) $ is described in the literature. It is $ W ^ {1} _ {g-1} = {l in the image ^ {g-1} (C): h ^ 0 (l) geq 2 } $.

I was able to find in the literature the expected results of the dimension.

Are the singularities of $ W ^ 1_ {g-1} $ known.

Question: What is the dimension (exp) of $ Sing (W ^ 1_ {g-1}) $, for a generic curve?

## python – Batch retrieves address formatted with geometry (lat / long) and output in csv format

I have a csv file with 3 fields, two of which are of my interest, `Merchant's name`

and `City`

.

My goal was to generate several csv files each containing 6 fields, `Merchant's name`

, `City`

, `first name`

, `formaté_adresse`

, `latitude`

, `longitude`

.

For example, if an entry in the csv file is `Starbucks, Chicago`

, I want the output csv to contain all the information in the 6 fields (as mentioned above) as`Starbucks, Chicago, Starbucks, "200 S Michigan Ave, Chicago, IL 60604, United States", 41.8164613, -87.8127855`

,`Starbucks, Chicago, Starbucks, "8 North Michigan Ave, Chicago, IL 60602, United States", 41.8164613, -87.8127855`

and so on for the rest of the results.

For this, I used the text search query of the Google Maps Places API. Here is what I wrote.

```
import pandas as pd
# import googlemaps
import requests
# import csv
# import pprint in pp
from the hour of import sleep
randomly import
def search_output (search):
if len (data['results']) == 0:
print (No results were found for {}. & # 39; format (search))
other:
# Create a csv file
filename = search + & # 39; .csv & # 39;
f = open (file name, "w")
size_of_json = len (data['results'])
# Get the token from the next page
# if size_of_json = 20:
# next_page = data['next_page_token']
for i in the range (size_of_json):
name = data['results'][i]['name']
address = data['results'][i]['formatted_address']
latitude = data['results'][i]['geometry']['location']['lat']
longitude = data['results'][i]['geometry']['location']['lng']
f.write (name.replace (& # 39;) & # 39;) + & # 39; + address.replace (& # 39;, & # 39;) ,, & # 39;) + & # 39;, & # 39; + str (latitude) + & # 39; + str (longitude) + & # 39; n & # 39; ;)
f.close ()
print (& # 39; File successfully saved for "{}". format. (search))
to sleep (random.randint (120, 150))
API_KEY = & # 39; your_key_here & # 39;
PLACES_URL = https://maps.googleapis.com/maps/api/place/textsearch/json? & # 39;
# Make dataframe
df = pd.read_csv (& # 39; merchant.csv & # 39 ;, usecols =[0, 1])
# Build a search query
search_query = df['Merchant_Name'].astype (str) + & # 39; & # 39; + df['City']
search_query = search_query.str.replace (& # 39 ;, & # 39; + & # 39;)
random.seed ()
for search in search_query:
search_req = & # 39; query = {} & key = {} & # 39; format (search, API_KEY)
request = PLACES_URL + search_req
# Place the request and store the data in & # 39; data & # 39;
result = requests.get (request)
data = result.json ()
status = data['status']
if the status == & # 39; OK & # 39 ;:
search_output (search)
status elif == & ZERO_RESULTS & # 39 ;:
print ("Zero results for" {} ". Moving on .. (format))
to sleep (random.randint (120, 150))
elif status == OVER_QUERY_LIMIT:
print (Limit the query reached! Try after a while. Can not complete the "{}". format. (search))
Pause
other:
print (status)
print (& # 39; ^ Status not okay, try again. Failed to complete "{}". format (search))
Pause
```

I want to implement the next page token but I can not think of a way that would not make everything mess. Another thing I want to improve is my CSV writing block. And dealing with redundancy.

I also plan to concatenate all csv files into one (while keeping the original separate files).

Please note that I am new to programming. In fact, it's actually one of my first programs to do something. So, please, elaborate a little more if need be. Thank you!

## Complex geometry – relatively early polynomials are relatively good as analytic germs

Let $ p, q $ are two relatively early polynomials in $ mathbb {C}[z_{1},ldots,z_{m}]$ that is, they have no common irreducible factor $ mathbb {C}[z_{1},ldots,z_{m}]$. Is it true that $ p_ {x}, q_ {x} $ are relatively first in $ mathcal {O} _ {x} $, for everyone $ x in Z (p) cap Z (q) $ $ ( gcd (p_ {x}, q_ {x}) $ is a unit of $ mathcal {O} _ {x} $?

## Asymptotic Geometry – Exponential decay of a holomorphic map at $ J $ on a long cylinder

assume $ (X, Omega, J) $ is a closed symplectic variety with an almost complex compatible structure. The following fact follows from McDuff-Salamon's book on $ J– holomorphic curves (more precisely, Lemma 4.7.3).

Given $ 0 < mu <1 $, there are constants $ 0 <C < infty $ and $ hbar> $ 0 such that the following property is valid. Given a $ Jholomorphic map $ u: (- R-1, R + 1) times S ^ 1 to X $ with energy $ E (u) < hbar $ defined on a ring (with $ R> $ 0), we have estimates of exponential decay

(1) $ E (u | _ {[-R+T,R-T] times S ^ 1}) C ^ 2e ^ {- 2 mu T} E (u) $

(2) $ sup_ {[-R+T,R-T] times S ^ 1} | of | ^ ^ – mu T} sqrt {E (u)} $

for everyone $ 0 the T the R $. Here we take $ S ^ 1 = mathbb R / 2 pi mathbb Z $ and use the standard flat metric on the cylinder $ mathbb R times S ^ 1 $ and the metric on $ X $ measure the norm $ | of | $.

Now if $ J were integrable, we can actually improve this estimate in the following way: to the detriment of $ hbar $ and increasing $ C $we can really take $ mu = 1 $ in (1) and (2) above. The idea would be to use (2) to deduce that $ u | _ {[-R,R] times S ^ 1} $ actually maps in a complex neighborhood of coordinates on $ X $ where we can use the Fourier expansion $ u $ along the cylinder $[-R,R] times S ^ 1 $ to get the desired estimate.

**I would like to know: is it possible to improve the estimate for $ mu = 1 $ also in the case where $ J is not integrable? If so, a proof with some details or a reference would be appreciated. If no, what is the reason and is it possible to find a (cons) example to illustrate this?**

## Non-commutative geometry – Equivalence of two approaches of transversal measurements for a foliation

Assume that $ (V, F) $ is a flaky variety. There are three approaches equivalent to the notion of *cross measurement* as described in this book (see pages 65-69). I would like to understand the last line of section 5$ alpha $ where it is stated that ,, it is easy to check, as in the case of flows, that $ Lambda $ meets the definition 2 & # 39 ;. The context is as follows: we start with a closed current $ C $ (of degree $ = dim F $) positive in the sense of the sheet and with the help of this current defines a measure $ mu_U $ (locally on $ U $, domain of the foliation sheet) on the plate set by the fromula

$$ langle C, omega rangle = int Big ( int _ { pi} omega Big) of mu_U ( pi). $$

Once we have this measurement, we can define $ Lambda (B): = int Map (B cap pi) of mu_ {U} ( pi) $ for any cross Borel $ B $ (ie Borel subassembly $ B subset V $ as for each sheet $ L $ of foliation $ B cap L $ is at most countable.

Why $ Lambda $ satisfied $ Lambda (B) = Lambda ( psi (B)) $ for any injection of Borel $ psi $ which keeps the leaves.

I guess that should somehow stem from the state of $ C $ closed, but I do not know how to perform the calculations (for example, a problem I encountered is that $ Lambda $ is defined locally and I do not see how to move from $ U $ to another table of foliation $ U $ (which can happen for the general $ psi $).

## dg.differential geometry – Book – Tetrahedron rolled on a plane

I'm looking for a book called "Charles W. Trigg." Tetrahedron rolled in a plane. J. Recreational Mathematics, 3 (2): 82-87, 1970. "

This is from @Joseph O 'Rourke comment in the previous post "Hamiltonian Cycles"

However, I have not seen this book anywhere on the Internet. Can any one please share the information of the book?

Thank you.

## Differential geometry – Riemann curvature and Levi-Civita connection on symmetric positive definite matrix collector

Let $ mathcal {S} $ to be the symmetric matrices and $ mathcal {P} $ to be positive defined matrices.

$ mathcal {S} $ naturally carries the structure of a vector space. Indoor product on $ mathcal {S} $ is given by $ langle A, B rangle = trace (A B) $ .

$ mathcal {P} $ is an open set in ( mathcal {S} $.

the map $$ log: mathcal {P} rightarrow mathcal {S} $$

is a diffeomorphism between two varieties. We can identify the tangent space in x $ T _ { text {x}} mathcal {P} $ with $ text {x} times mathcal {S} $.

The metric induced on the Tangent space is given by $$ langle A, B rangle_ {x} = langle dlog (A) | _ {x}, dlog (B) | _ {x} rangle $$ ,or

$$

dlog: T_x mathcal {P} rightarrow T_x mathcal {S} $$

$$

An mapsto an X ^ {- 1}

$$

Explicitly write the inner product on $ mathcal {P} $ is given by

$$ langle A, B rangle_ {X} = text {trace} (A X ^ {- 1} B X ^ {- 1}) $$

Length $ L ( gamma) $ and energy $ E ( gamma) $ of a curve $ gamma: [0,1] rightarrow mathcal {P} $ is given by

$$

L ( gamma) = int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} dt $$

$$ E ( gamma) = frac {1} {2} int_a ^ b | dfrac {d gamma} {dt} | _ { gamma (t)} ^ 2 dt $$

Geodesics are curves that minimize energy on a collector. They allow us to introduce a distance between two points of a manifold. To calculate them we can simply use euler's lagler equations

$$ frac {d} {d}} frac {d} {d dot { gamma}} f (t, gamma (t), dot { gamma} (t)) = frac {d} {d gamma} f (t, gamma (t), point { gamma} (t)) $$

write explicitly we have

$$ frac {d} {dt} frac {d} {d dot { gamma}}

text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}) = frac {d} {d gamma}

text {trace} ( dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1}) $$

The left side of the equation reduces to

$$ frac {d} {d}} gamma ^ {- 1} dot { gamma} gamma ^ {- 1} $$

while the right side of the equation reduces to

$$ – gamma ^ {- 1} dot { gamma} gamma ^ {- 1} dot { gamma} gamma ^ {- 1} $$

By solving the differential equation, we obtain two different expressions for geodesics.

$$ gamma (t) = P exp (t P ^ {- 1} S) $$

Here, P is located on the variety and S is located on the tangent space. intuitively $ gamma $ is the curve that starts in P with the direction S.

$$ gamma_ {AB} (t) = A (A ^ {- 1} B) ^ t $$

Here, gamma is the geodesic between the points of variety A and B.

Now that the geodesics are defined, the distance on the spd manifold can be defined via the length of the geodesic curve connecting two points. After some calculations we come to

$$ d (X, Y) = | log (X ^ {- 1} Y) | $$

So, these are my calculations up to now. I am pretty new to differential geometry. My first question is: Do arguments usually make sense? I'm not really convinced that $ dlog = dX X ^ {- 1} $ or that the internal product on $ mathcal {S} $ is $ trace (AB) $. My ultimate goal is to read the christoffel symbols from the geodesic eq. and calculate the riem. curvature. But I'm not sure how to proceed or if calculations have up to now a meaning

## dg.differential geometry – Lie transformation group and transformation of the smooth structure from a normal connected subgroup

I am working on two theorems about the Lie transformation group of the Kobayashi book transformation group in differential geometry, one of which is as follows:

TheoremLet $ mathfrak {S} $ the differentiable variety transformation group $ M $ and $ mathcal {S} $ be the set of all vector fields $ X in mathfrak {X} (M) $ which generates a subgroup of 1 parameter $ varphi_ {t} = text {exp} (tX) $

of transformation for $ M $ such as $ varphi_ {t} in mathfrak {S} $. L & # 39; together $ mathcal {S} $ with vector field brackets define a Lie algebra. Then if $ mathcal {S} $ is the finite dimensional Lie algebra of vector fields on $ M $ then $ mathfrak {S} $ is the Lie transformation group and $ mathcal {S} $ his Lie algebra.

the idea of the proof is quite simple, let's take the Lie algebra $ mathfrak {g} ^ {*} $ generated by $ mathcal {S} $, if $ mathfrak {g} ^ {*} $ if the third Lie theorem is of finite dimension, there is simply a Lie group $ mathfrak {S} ^ {*} $ the last one can be chosen as $ mathfrak {S} ^ {*} subset mathfrak {S} $ using local action, etc …, we can further show that the connected subgroup is normal and open, for the moment, everything seems normal, but the author claims that the smooth structure can be transformed into a other component connected or just tell to $ mathfrak {S} $ if I'm not mistaken, I can not get it unless I accept that the left translation or the correct one $ g in mathfrak {S} $, cartography $ L_ {g}: mathfrak {S} ^ {*} longrightarrow g. Mathfrak {S} ^ {*} $ to be differentiable.

My question concerns the idea of how to transfer a smooth structure from a normal connected subgroup?

There is an article by Richard S.Plais A comprehensive formulation of the transformation group lie theory contains a lot of things about it, but I could not see exactly where to find the answer.