Revisions to Polynomial contact structures on $RP^3$

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edited Mar 28, 2015 at 21:52

Nikita Kalinin

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Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2\ $$f2=x^4+y^4+z^4+t^4\ $ (in the above notation) does NOT produce contact structure. IsIt is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2\ $ (in the above notation) does NOT produce contact structure. Is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^4+y^4+z^4+t^4\ $ (in the above notation) does NOT produce contact structure. It is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

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edited Feb 27, 2015 at 11:43

Nikita Kalinin

5k
1
40
58

Here is a code in MacaulayMacaulay2 for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2\ $ DOES not(in the above notation) does NOT produce contact structure. Is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2\ $ DOES not produce contact structure. Is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2\ $ (in the above notation) does NOT produce contact structure. Is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why the induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

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edited Apr 2, 2012 at 9:24

Nikita Kalinin

5k
1
40
58

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure

R = QQ[x,y,z,t]

degf=4

f=x^degf+y^degf+z^degf+t^degf + 3x^2z^2 + x^2*t^2

--f2 = x^degf+y^degf+z^degf+t^degf

nR = diff(y,f)*x-diff(x,f)*y+diff(t,f)*z-diff(z,f)*t

cdx = degffdiff(y,f)-nR*diff(x,f)

cdy = -degffdiff(x,f)-nR*diff(y,f)

cdz = degffdiff(t,f)-nR*diff(z,f)

cdt = -degffdiff(z,f)-nR*diff(t,f)

cxyz=cdx*(diff(y,cdz)-diff(z,cdy))+cdy*(diff(z,cdx)-diff(x,cdz))+cdz*(diff(x,cdy)-diff(y,cdx))

re = sub(cxyz,t=>1)

factor re

So, $f2=x^2+y^2+z^2+t^2$$f2=x^2+y^2+z^2+t^2\ $ DOES not produce contact structure. Is convex but not strictly(!) plurisubharmonic (on the plane $x=y=0$). Here (http://www.math.ethz.ch/~evansj/lecture9.pdf) there is a good explanation why induced structure is contact ($d\eta$ tames complex structure on $\mathbb R^4$ )

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edited Mar 6, 2012 at 11:54

Nikita Kalinin

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created Nov 12, 2011 at 0:02

Nikita Kalinin

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Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

Here is a code in MacaulayMacaulay2 for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay2 for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure

Here is a code in Macaulay for checking that some plurisubharmonic function determines a contact structure