On the classification of Kähler-Ricci solitons on Gorenstein del Pezzo surfaces: Calculations for threefolds
Here some helper functions are defined. These are degenerations()
for calculating the toric special fibres of the test configurations and intxexp()
to solve the integrals analytically.
Click ▸ on the left to unhide.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every (admissible) choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every (admissible) choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every (admissible) choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this we have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
We identify a small closed rectangle containing our estimate such that for any outer normal of this rectangle, where . This and uniqueness guarantee our candidate lies within the rectangle.
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every (admissible) choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence, we have We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every choice of and then plug in the estimate for into the resulting expression.
Step (i) -- obtain a closed form for
For this note, that is a symmetry of the combinatorial data. Hence we have . We have to analytically solve the integral for varying over a basis of .
Step (ii) -- find an estimate for the soliton candidate vector field .
Step (iii) & (iv) -- obtain closed forms for and plug in the estimate for
For this we first have to symbolically solve the integrals for every choice of and then plug in the estimate for into the resulting expression.